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cy-liczby-pierwsze",children:"Wielomian Eulera generujÄ
cy liczby pierwsze"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#spirala-ulama",children:"Spirala Ulama"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#funkcja-zliczajÄ
ca-liczby-pierwsze",children:"Funkcja zliczajÄ
ca liczby pierwsze"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#twierdzenie-o-liczbach-pierwszych",children:"Twierdzenie o liczbach pierwszych"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#liczby-pierwsze-mersennea",children:"Liczby pierwsze Mersenne'a"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#inne-warte-uwagi-rodzaje-liczb-pierwszych",children:"Inne warte uwagi rodzaje liczb pierwszych"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#maÅe-twierdzenie-fermata",children:"MaÅe twierdzenie Fermata"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#liczby-pseudopierwsze",children:"Liczby pseudopierwsze"})}),(0,n.jsxs)(e.li,{className:"toc-item toc-item-h2",children:[(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#testowanie-dużych-liczb-pierwszych",children:"Testowanie dużych liczb pierwszych"}),(0,n.jsxs)(e.ol,{className:"toc-level toc-level-2",children:[(0,n.jsxs)(e.li,{className:"toc-item toc-item-h3",children:[(0,n.jsx)(e.a,{className:"toc-link toc-link-h3",href:"#silne-testy-pseudopierwszoÅci",children:"(Silne) testy pseudopierwszoÅci"}),(0,n.jsxs)(e.ol,{className:"toc-level toc-level-3",children:[(0,n.jsx)(e.li,{className:"toc-item toc-item-h4",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h4",href:"#test-pierwszoÅci-fermata",children:"Test pierwszoÅci Fermata"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h4",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h4",href:"#test-millera-rabina",children:"Test Millera-Rabina"})})]})]}),(0,n.jsxs)(e.li,{className:"toc-item toc-item-h3",children:[(0,n.jsx)(e.a,{className:"toc-link toc-link-h3",href:"#klasyczne-testy",children:"Klasyczne testy"}),(0,n.jsxs)(e.ol,{className:"toc-level toc-level-3",children:[(0,n.jsx)(e.li,{className:"toc-item toc-item-h4",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h4",href:"#n---1",children:"n - 1"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h4",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h4",href:"#n--1",children:"n + 1"})})]})]}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h3",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h3",href:"#testy-og\xf3lnego-przeznaczenia",children:"Testy og\xf3lnego przeznaczenia"})})]})]}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#podsumowanie",children:"Podsumowanie"})}),(0,n.jsx)(e.li,{className:"toc-item toc-item-h2",children:(0,n.jsx)(e.a,{className:"toc-link toc-link-h2",href:"#literatura",children:"Literatura"})})]})}),"\n","\n",(0,n.jsxs)(e.h2,{id:"matematyka-liczb-pierwszych",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#matematyka-liczb-pierwszych",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Matematyka liczb pierwszych"]}),"\n",(0,n.jsx)(e.p,{children:"Matematycy od dawnych lat pr\xf3bowali dojÅÄ do tego, jak otrzymywaÄ liczby pierwsze proÅciej niż iteracyjnymi sposobami takimi jak sito Eratostenesa. Z jednej strony w ramach ich zainteresowania byÅ temat, jak szybciej sprawdzaÄ
1 pierwszoÅÄ zamiast dzielenia po kolei aż do pierwiastka z liczby. Z drugiej strony chcieli odkryÄ, czy istnieje coÅ w stylu wzoru na liczbÄ pierwszÄ
. Nawet niekoniecznie wz\xf3r dajÄ
cy zawsze liczbÄ pierwszÄ
, co jakakolwiek zależnoÅÄ pozwalajÄ
ca Åatwiej namierzyÄ liczby pierwsze w nieskoÅczonym zbiorze liczb naturalnych; czy nawet jakakolwiek wÅasnoÅÄ wsp\xf3lna dla wszystkich liczb pierwszych, kt\xf3ra uÅatwi ich znalezienie."}),"\n",(0,n.jsxs)(e.p,{children:["Szczeg\xf3lnie staÅo siÄ to istotne dziÅ, gdy uzależniamy dziaÅanie wielu ważnych algorytm\xf3w (np. RSA) od wygenerowania dużych liczb pierwszych. PrzykÅadowo, jeÅli chcielibyÅmy wygenerowaÄ liczbÄ pierwszÄ
zajmujÄ
cÄ
2048 bit\xf3w, to mamy do sprawdzenia ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"2046"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"8"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"615"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^{2046} \\approx 8 \\cdot 10^{615}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"2046"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"8"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"615"})})})]})})})})})]})]})]})]})," liczb (pierwszy bit musi byÄ 1, żeby liczba byÅa tej dÅugoÅci, a ostatni r\xf3wnież 1, aby byÅa nieparzysta). Każda z nich bÄdzie miaÅa w okolicach 616 cyfr, wiÄc sprawdzenie metodÄ
naiwnÄ
bÄdzie dÅugotrwaÅe. Sita r\xf3wnież niekoniecznie siÄ sprawdzÄ
, bo każda z liczb bÄdzie zajmowaÄ 256 bajt\xf3w w pamiÄci, wiÄc sprawdzenie wszystkich zajÄÅoby mniej wiÄcej ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"606"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"10^{606}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"606"})})})]})})})})})]})]})})]})," terabajt\xf3w."]}),"\n",(0,n.jsx)(e.p,{children:"Dziedzina ta jest bardzo szeroka, bo i przez wiele lat udaÅo siÄ matematykom odkryÄ wiele ciekawych zależnoÅci, choÄ nie zawsze zostaÅy do koÅca udowodnione. W artykule pokaÅ¼Ä moim zdaniem najważniejsze i najciekawsze z nich."}),"\n",(0,n.jsxs)(e.h2,{id:"wielomian-eulera-generujÄ
cy-liczby-pierwsze",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#wielomian-eulera-generujÄ
cy-liczby-pierwsze",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Wielomian Eulera generujÄ
cy liczby pierwsze"]}),"\n",(0,n.jsx)(e.p,{children:"NaszÄ
matematycznÄ
przygodÄ zaczniemy od odkrycia Eulera z 1772 r., kt\xf3ry zauważyÅ, że wielomian:"}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"41"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(n) = n^2 + n + 41"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.9474em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8641em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"41"})]})]})]})}),"\n",(0,n.jsxs)(e.p,{children:["generuje liczby pierwsze dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"0"}),(0,n.jsx)(e.mo,{children:"⩽"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"⩽"}),(0,n.jsx)(e.mn,{children:"39"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"0 \\leqslant n \\leqslant 39"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7811em",verticalAlign:"-0.1367em"}}),(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel amsrm",children:"⩽"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7
1733em",verticalAlign:"-0.1367em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel amsrm",children:"⩽"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"39"})]})]})]}),". Nie uzyskamy w ten spos\xf3b dużych liczb, bo najwiÄkszÄ
jest 1601. Do tego nie sÄ
to też wszystkie po kolei. Jednak co warto tutaj zobaczyÄ, to dalsze odkrycia. Mianowicie zauważono, że nie tylko 41 tak siÄ zachowuje. JeÅli wz\xf3r zgeneralizujemy do:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mi,{children:"p"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(n) = n^2+n+p"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.9474em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord 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mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})]})]})}),"\n",(0,n.jsxs)(e.p,{children:["to okazuje siÄ, że liczby pierwsze bÄdziemy w stanie generowaÄ dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"3"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"5"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"11"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"17"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"41"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"}"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p \\in \\{1, 2, 3, 5, 11, 17, 41\\}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7335em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord 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1),". SÄ
to tak zwane szczÄÅliwe liczby Eulera."]}),"\n",(0,n.jsxs)(e.p,{children:["Co wiÄcej, wzory te r\xf3wnież potrafiÄ
dalej generowaÄ liczby pierwsze, tylko pojawiÄ
siÄ co jakiÅ czas liczby zÅożone. Liczby zÅożone otrzymamy zawsze, jeÅli ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," jest podzielne przez ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"p"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})})]})," oraz ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"(n+1)"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})]})]})," jest podzielne przez ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"p"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})})]}),"."]}),"\n",(0,n.jsxs)(e.h2,{id:"spirala-ulama",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#spirala-ulama",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Spirala Ulama"]}),"\n",(0,n.jsx)(e.p,{children:"W 1963 r. polski matematyk StanisÅaw Ulam zaproponowaÅ nowy spos\xf3b wizualizacji liczb pierwszych, kt\xf3ry ujawniÅ ich wciÄ
ż niewyjaÅnione wÅaÅciwoÅci. Polega ona na rozpisaniu po kolei wszystkich liczb naturalnych od 1 wzwyż na spirali, a nastÄpnie zaznaczeniu liczb pierwszych tak jak na rysunku poniżej."}),"\n",(0,n.jsxs)(t.xA,{minWidth:266,children:[(0,n.jsx)(t._V,{image:p,alignCenter:!0,fullSize:!0,alt:"Spirala z rozpisanymi wszystkimi liczbami od 1 do 49",caption:(0,n.jsxs)(n.Fragment,{children:["Spirala z rozpisanymi wszystkimi liczbami od 1 do 49.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Onmywaybackhome - Own work, Public Domain, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=8327264",target:"_blank",children:"https://commons.wikimedia.org/w/index.php?curid=8327264"}),")"]}
1),forceWhiteBackground:!0}),(0,n.jsx)(t._V,{image:j,alignCenter:!0,fullSize:!0,alt:"Spirala z rozpisanymi liczbami pierwszymi od 1 do 49",caption:(0,n.jsxs)(n.Fragment,{children:["Spirala z zaznaczonymi liczbami pierwszymi.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Onmywaybackhome - Own work, Public Domain, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=8327308",target:"_blank",children:"https://commons.wikimedia.org/w/index.php?curid=8327308"}),")"]}),forceWhiteBackground:!0})]}),"\n",(0,n.jsx)(e.p,{children:"Na takiej maÅej iloÅci liczb nie widaÄ jeszcze żadnych zależnoÅci, mimo że sam ksztaÅt już zaczyna byÄ ciekawy. Zobaczmy spiralÄ Ulama dla wiÄkszej iloÅci liczb:"}),"\n",(0,n.jsx)(t._V,{image:d,alignCenter:!0,fullSize:!0,alt:"Spirala Ulama dla 40 tysiÄcy liczb",caption:(0,n.jsxs)(n.Fragment,{children:["Spirala Ulama dla 40 tysiÄcy liczb.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Grontesca at the English Wikipedia, CC BY-SA 3.0, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=1924394",target:"_blank",children:"https://commons.wikimedia.org/w/index.php?curid=1924394"}),")"]})}),"\n",(0,n.jsxs)(e.p,{children:["To, co zainteresowaÅo matematyk\xf3w, to linie, w jakie ukÅadajÄ
siÄ niekt\xf3re z liczb pierwszych. PatrzÄ
c na ten rysunek, możemy zauważyÄ, że ewidentnie jest jakaÅ zależnoÅÄ ukrywajÄ
ca siÄ za rozmieszczeniem liczb pierwszych, tylko wciÄ
ż nie wiemy, jaka dokÅadnie. Jednak jesteÅmy w stanie uzyskaÄ wzory na niekt\xf3re z tych linii. Na przykÅad na poniższym rysunku zaznaczono liczby pierwsze generowane przez wz\xf3r ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"4"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"41"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"4x^2-2x+41"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord",children:"4"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"41"})]})]})]})," (podobny do wzoru Eulera, czyż nie?):"]}),"\n",(0,n.jsx)(t._V,{image:x,alignCenter:!0,fullSize:!0,alt:"Spirala Ulama dla 40 tysiÄcy liczb z zaznaczonymi liczbami speÅniajÄ
cymi wz\xf3r 4x^2-2x+41",caption:(0,n.jsxs)(n.Fragment,{children:["Na granatowo zaznaczono liczby speÅniajÄ
ce ten wz\xf3r dla x ze zbioru liczb naturalnych. Wyr\xf3żniajÄ
ca siÄ, r\xf3wnolegÅa linia na dole rysunku jest możliwa do uzyskania tym samym wzorem, ale z ujemnymi wartoÅciami x.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Will Orrick - Own work, Public Domain, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=10720576",children:"https://commons.wikimedia.org/w/index.php?curid=10720576"}),")"]})}),"\n",(0,n.jsxs)(e.h2,{id:"funkcja-zliczajÄ
ca-liczby-pierwsze",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#funkcja-zliczajÄ
ca-liczby-pierwsze",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Funkcja zliczajÄ
ca liczby pierwsze"]}),"\n",(0,n.jsxs)(e.p,{children:["Poza odnalezieniem wzoru, kt\xf3ry dawaÅby nam liczby pierwsze, matematycy skupiajÄ
siÄ też na znalezieniu funkcji zliczajÄ
cej liczby pierwsze. Możesz teraz zadaÄ pytanie: tylko po co? Odpowiadam: choÄby dziÄki temu bylibyÅmy w stanie okreÅliÄ prawdopodobieÅstwo znalezienia liczby pierwszej w pewnym skoÅczonym zbiorze liczb naturalnych. FunkcjÄ takÄ
tradycyjnie zapisuje siÄ jako ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pi(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"Ï"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]}),"."]}),"\n",(0,n.jsx)(t._V,{image:h,alignCenter:!0,fullSize:!0,alt:"Wykres wartoÅci funkcji pi(n) w zależnoÅci od wartoÅci n",caption:(0,n.jsxs)(n.Fragment,{children:["WartoÅci funkcji zliczajÄ
cej liczby pierwsze dla 60 pierwszych liczb naturalnych.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Bender2k14 - Mathematica source code, CC BY-SA 3.0, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=17040316",target:"_blank",children:"https://commons.wikimedia.org/w/index.php?curid=17040316"}),")"]}
1),forceWhiteBackground:!0}),"\n",(0,n.jsxs)(e.p,{children:["Prawdopodobnie nie zaskoczÄ nikogo faktem, że nie znamy dokÅadnego wzoru. Jednak z racji tego, że znamy iloÅÄ liczb pierwszych aż do ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"28"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"10^{28}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"28"})})})]})})})})})]})]})})]})," (Baugh D., Walisch K., 2020), jesteÅmy w stanie oszacowaÄ wz\xf3r na r\xf3żne sposoby, a także okreÅliÄ jego bÅÄ
d."]}),"\n",(0,n.jsx)(e.p,{children:"NajczÄÅciej spotykanym przybliżeniem tej funkcji jest wz\xf3r zaproponowany pod koniec XVIII w. przez Gaussa i Legendre, wskazujÄ
cy jej asymptotyczne tempo wzrostu*:"}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"â¼"}),(0,n.jsxs)(e.mfrac,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"log"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pi(x) \\sim \\frac{x}{\\log(x)}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"Ï"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â¼"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"2.0436em",verticalAlign:"-0.936em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,n.jsx)(e.span,{className:"mfrac",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"1.1076em"},children:[(0,n.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsxs)(e.span,{className:"mop",children:["lo",(0,n.jsx)(e.span,{style:{marginRight:"0.01389em"},children:"g"})]}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})]}
1),(0,n.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,n.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"})})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.936em"},children:(0,n.jsx)(e.span,{})})})]})}),(0,n.jsx)(e.span,{className:"mclose nulldelimiter"})]})]})]})]})}),"\n",(0,n.jsxs)(e.p,{children:["BÅÄ
d wzglÄdny tego wzoru dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"100"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"x=100"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"100"})]})]})]})," wynosi 13,2 %, jednak wraz ze wzrostem ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"x"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"})]})})]})," bÅÄ
d spada. Dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"28"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"x=10^{28}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"28"})})})]})})})})})]})]})]})]})," (najwyższe, dla kt\xf3rego znamy dokÅadnÄ
wartoÅÄ) bÅÄ
d wynosi już zaledwie 1,58 %. SwojÄ
drogÄ
, jeÅli chcemy wyliczaÄ z tego wzoru przybliżone wartoÅci ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pi(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"Ï"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]}),", to należy uznaÄ, że jest to logarytm naturalny (mimo zapisu sugerujÄ
cego podstawÄ 10)."]}),"\n",(0,n.jsxs)(e.p,{children:["Innym znanym przybliżeniem jest wartoÅÄ logarytmu caÅkowego z ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"x"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"})]})})]}),":"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mtable,{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em",children:[(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mrow,{}),(0,n.jsx)(e.mo,{children:"â¼"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"Li"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{separator:"true",children:","})]})})})]}),(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsx)(e.mrow,{})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mrow,{}),(0,n.jsx)(e.mtext,{children:"gdzie:\xa0"})]})})})]}),(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"Li"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mrow,{}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msubsup,{children:[(0,n.jsx)(e.mo,{children:"â«"}),(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"x"})]}),(0,n.jsxs)(e.mfrac,{children:[(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"ln"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mi,{children:"t"})]})]}),(0,n.jsx)(e.mi,{children:"d"}),(0,n.jsx)(e.mi,{children:"t"})]})})})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\begin{align*}\n\\pi(x) &\\sim \\operatorname{Li}(x), \\\\ &\\text{gdzie: } \\\\\n\\operatorname{Li}(x) &= \\int_2^x \\frac{1}{\\ln t}dt\n\\end{align*}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"5.6262em",verticalAlign:"-2.5631em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsxs)(e.span,{className:"mtable",children:[(0,n.jsx)(e.span,{className:"col-align-r",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"3.0631em"}
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1,children:[(0,n.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mop",children:"ln"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})]}),(0,n.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,n.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord",children:"1"})})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.686em"},children:(0,n.jsx)(e.span,{})})})]})}),(0,n.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"d"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"2.5631em"},children:(0,n.jsx)(e.span,{})})})]})})]})})]})})]})}),"\n",(0,n.jsxs)(e.p,{children:["* Asymptotyczne tempo wzrostu okreÅla zachowanie wartoÅci funkcji wraz ze wzrostem argument\xf3w. Innymi sÅowy, wskazuje âmniej wiÄcejâ ksztaÅt wykresu funkcji, jednak nie jej dokÅadne wartoÅci. Na pewno znasz to z informatyki, ale pod innÄ
nazwÄ
â notacja dużego ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{mathvariant:"normal",children:"O"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\Omicron"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,n.jsx)(e.span,{className:"mord mathrm",children:"O"})]})})]})," (ciekawostka: wbrew powszechnemu przekonaniu nie jest to litera O, tylko duża grecka litera omikron)."]}),"\n",(0,n.jsxs)(e.h2,{id:"twierdzenie-o-liczbach-pierwszych",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#twierdzenie-o-liczbach-pierwszych",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Twierdzenie o liczbach pierwszych"]}),"\n",(0,n.jsxs)(e.p,{children:["Na bazie funkcji zliczajÄ
cej liczby pierwsze możliwe byÅo sformalizowanie spostrzeżenia, że liczby pierwsze sÄ
coraz rzadsze, im dalej od zera siÄ znajdujemy. WÅaÅnie twierdzenie o liczbach pierwszych stwierdza, że ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.mfrac,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"log"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mi,{children:"x"})]})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\frac{x}{\\log{x}}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1.1765em",verticalAlign:"-0.4811em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,n.jsx)(e.span,{className:"mfrac",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"0.6954em"},children:[(0,n.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsxs)(e.span,{className:"mop mtight",children:[(0,n.jsx)(e.span,{className:"mtight",children:"l"}),(0,n.jsx)(e.span,{className:"mtight",children:"o"}),(0,n.jsx)(e.span,{className:"mtight",style:{marginRight:"0.01389em"},children:"g"})]}),(0,n.jsx)(e.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})})]})})]}
1),(0,n.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,n.jsxs)(e.span,{style:{top:"-3.394em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})})})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.4811em"},children:(0,n.jsx)(e.span,{})})})]})}),(0,n.jsx)(e.span,{className:"mclose nulldelimiter"})]})]})})]})," jest dobrym przybliżeniem ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pi(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"Ï"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]}),". Wiemy to, ponieważ wraz ze wzrostem wartoÅci x, stosunek wÅaÅciwej wartoÅci funkcji oraz jej przybliżenia zbiega do 1, co matematycznie wyrazimy jako:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.munder,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"lim"}),(0,n.jsx)(e.mo,{children:"â¡"})]}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"â"})]})]}),(0,n.jsxs)(e.mfrac,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Ï"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsxs)(e.mfrac,{children:[(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"log"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})]})]}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\lim_{x\\to\\infty}\\frac{\\pi(x)}{\\frac{x}{\\log(x)}} = 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"2.633em",verticalAlign:"-1.206em"}}),(0,n.jsx)(e.span,{className:"mop op-limits",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"0.6944em"},children:[(0,n.jsxs)(e.span,{style:{top:"-2.4em",marginLeft:"0em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"}),(0,n.jsx)(e.span,{className:"mrel mtight",children:"â"}),(0,n.jsx)(e.span,{className:"mord mtight",children:"â"})]})})]}
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cy bÅÄ
d przybliżeÅ funkcji pi(x) wzglÄdem jej prawdziwych wartoÅci",caption:(0,n.jsxs)(n.Fragment,{children:["Wykres pokazuje, jak zachowujÄ
siÄ wartoÅci obu iloraz\xf3w wraz ze wzrostem x (oÅ pozioma, skala logarytmiczna
1). Widzimy z tego, że pierwszy wz\xf3r (niebieska linia) przybliża wartoÅÄ z niedomiarem i powoli zbiega do 1. Drugi wz\xf3r (czerwona linia) natomiast przybliża z nadmiarem, jednak wartoÅÄ szybciej zbiega do 1.",(0,n.jsx)(e.br,{}),"(źr\xf3dÅo: By Dcoetzee - Own work, CC0, ",(0,n.jsx)(e.a,{href:"https://commons.wikimedia.org/w/index.php?curid=25216710",target:"_blank",children:"https://commons.wikimedia.org/w/index.php?curid=25216710"}),")"]}),forceWhiteBackground:!0}),"\n",(0,n.jsxs)(e.p,{children:["Co nam to daje w praktyce w kontekÅcie szukania liczb pierwszych? Dwie rzeczy. Po pierwsze, możemy okreÅliÄ, że n-ta liczba pierwsza (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mi,{children:"n"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p_n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]})]})})]}),") speÅnia nastÄpujÄ
cÄ
zależnoÅÄ:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"â¼"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mi,{children:"log"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p_n \\sim n \\log{n}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â¼"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsxs)(e.span,{className:"mop",children:["lo",(0,n.jsx)(e.span,{style:{marginRight:"0.01389em"},children:"g"})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})})]})]})]})}),"\n",(0,n.jsxs)(e.p,{children:["PrzykÅadowo, ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"17"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2\\cdot 10^{17}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"17"})})})]})})})})})]})]})]})]})," liczba pierwsza to 8 512 677 386 048 191 063, gdy z powyższego wzoru otrzymamy 7 967 418 752 291 744 388. BÅÄ
d wzglÄdny wynosi w tym przypadku okoÅo 6,4 %."]}),"\n",(0,n.jsxs)(e.p,{children:["Druga interesujÄ
ca rzecz â z twierdzenia tego wynika, że dla dużych ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," prawdopodobieÅstwo, że trafimy na liczbÄ pierwszÄ
nie wiÄkszÄ
od ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]}),", wynosi okoÅo ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.mfrac,{children:[(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"ln"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mi,{children:"n"})]})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\frac{1}{\\ln{n}}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,n.jsx)(e.span,{className:"mfrac",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"0.8451em"},children:[(0,n.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsxs)(e.span,{className:"mop mtight",children:[(0,n.jsx)(e.span,{className:"mtight",children:"l"}),(0,n.jsx)(e.span,{className:"mtight",children:"n"})]}),(0,n.jsx)(e.span,{className:"mspace mtight",style:{marginRight:"0.1952em"}}),(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})})]}),(0,n.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,n.jsxs)(e.span,{style:{top:"-3.394em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"1"})})})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.345em"},children:(0,n.jsx)(e.span,{})})})]})}),(0,n.jsx)(e.span,{className:"mclose nulldelimiter"})]})]})})]}),". IdÄ
c dalej tym tropem, szansa, że losowa liczba naturalna majÄ
ca co najmniej ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]}
1)," cyfr jest pierwsza, jest o poÅowÄ mniejsza niż dla liczby majÄ
cej ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," cyfr. Wynika to z twierdzenia, że Åredni odstÄp miÄdzy liczbami pierwszymi wynosi okoÅo ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"log"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\log{n}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsxs)(e.span,{className:"mop",children:["lo",(0,n.jsx)(e.span,{style:{marginRight:"0.01389em"},children:"g"})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})})]})})]}),"."]}),"\n",(0,n.jsxs)(e.p,{children:["Aby przeÅożyÄ te prawdopodobieÅstwa na konkretny przykÅad, wyobraźmy sobie, że chcemy wylosowaÄ liczbÄ pierwszÄ
, kt\xf3ra ma 1024 bity, czyli najwiÄksza wynosi ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"1024"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^{1024}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"1024"})})})]})})})})})]})]})})]}),". Oznacza to, że powinniÅmy siÄ spodziewaÄ, że aby znaleÅºÄ liczbÄ pierwszÄ
, wystarczy wylosowaÄ okoÅo 710 1024-bitowych liczb. Co wiÄcej, z racji tego, że możemy z g\xf3ry odrzuciÄ liczby parzyste, to możemy tÄ wartoÅÄ obniżyÄ o poÅowÄ. Sprawdzenie 355 liczb zamiast ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"1024"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"8"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"308"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^{1024} \\approx 1,8 \\cdot 10^{308}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"1024"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8389em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"8"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"308"})})})]})})})})})]})]})]})]})," brzmi jak coÅ o wiele bardziej osiÄ
galnego."]}),"\n",(0,n.jsxs)(e.h2,{id:"liczby-pierwsze-mersennea",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#liczby-pierwsze-mersennea",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Liczby pierwsze Mersenne'a"]}),"\n",(0,n.jsx)(e.p,{children:"Jednym z najbardziej znanych i rozpoznawalnych wzor\xf3w dajÄ
cych liczby pierwsze jest wz\xf3r na liczby Mersenne'a, kt\xf3re sÄ
tak nazwane po XVII-wiecznym francuskim matematyku Marinie Mersenne zajmujÄ
cym siÄ ich badaniem. Mimo że wz\xf3r ten nie daje wszystkich liczb pierwszych, nawet nie daje ich czÄsto, to jednak na pewno siÄ kiedykolwiek z nim spotkaÅeÅ(-aÅ). Ten wz\xf3r to:"}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"M"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_n = 2^n - 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7977em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.7144em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})}),"\n",(0,n.jsx)(e.p,{children:"Innymi sÅowy, sÄ
to liczby pierwsze o jeden mniejsze od potÄgi dw\xf3jki. 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1),(0,n.jsxs)(e.span,{style:{top:"-0.0997em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8641em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"13"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mord",children:"8192"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mord",children:"8191"})]})]})]}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"3.5603em"},children:(0,n.jsx)(e.span,{})})})]})})]})})]})})]})}),"\n",(0,n.jsxs)(e.p,{children:["Obecnie (luty 2022) znamy tylko 51 wartoÅci ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]}),", kt\xf3re dajÄ
nam liczby pierwsze. Pytaniem jest jednak, czy takich liczb jest nieskoÅczenie wiele? Jest to jeden z nierozwiÄ
zanych problem\xf3w matematycznych."]}),"\n",(0,n.jsxs)(e.p,{children:["Jednak jednoczeÅnie to wÅaÅnie liczby Mersenne'a sÄ
najwiÄkszymi z liczb pierwszych, jakie znamy. R\xf3wnież sÄ
stosowane w informatyce (patrz np. algorytm Mersenne Twister). Dlaczego tak jest? Ot\xf3ż sÄ
one najprostsze do przedstawienia ",(0,n.jsx)(e.a,{href:"/post/matematyka-zero-jedynkowa",children:"w formie binarnej"}),". Każda liczba ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^n-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7477em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.6644em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})," to w zapisie binarnym ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," jedynek. Sprawia to też, że obliczenia na nich sÄ
stosunkowo proste."]}),"\n",(0,n.jsxs)(e.p,{children:["Obecnie wciÄ
ż trwajÄ
poszukiwania coraz wiÄkszych liczb pierwszych speÅniajÄ
cych wÅaÅnie wz\xf3r Mersenne'a. Od 1996 r. dziaÅa program ",(0,n.jsx)(e.a,{href:"https://www.mersenne.org/",children:"GIMPS"})," (Great Internet Mersenne Prime Search), kt\xf3ry skupia siÄ na znajdowaniu jak najwiÄkszych liczb pierwszych Mersenne'a przez rozproszenie obliczeÅ na wiele komputer\xf3w. Każdy może udostÄpniÄ sw\xf3j komputer i wspom\xf3c znajdowanie kolejnych dużych liczb pierwszych."]}),"\n",(0,n.jsxs)(e.h2,{id:"inne-warte-uwagi-rodzaje-liczb-pierwszych",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#inne-warte-uwagi-rodzaje-liczb-pierwszych",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Inne warte uwagi rodzaje liczb pierwszych"]}),"\n",(0,n.jsx)(e.p,{children:"Liczby pierwsze Mersenne'a sÄ
najbardziej znane, ale mamy też inne rodzaje liczb pierwszych, o kt\xf3rych warto wspomnieÄ gÅ\xf3wnie ze wzglÄdu na to, że znamy sztuczki na ich w miarÄ proste testowanie:"}),"\n",(0,n.jsxs)(e.ul,{children:["\n",(0,n.jsxs)(e.li,{children:["Liczby pierwsze Fermata â jest to podzbi\xf3r liczb Fermata zawierajÄ
cy tylko liczby pierwsze. Wz\xf3r na liczbÄ Fermata to: ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"F"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"n"})]})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"F_n = 2^{2^n}+1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"F"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.9633em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.88em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mtight",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.7385em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.5em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size3 size1 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),". Liczby pierwsze znajdziemy dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,n.jsx)(e.mn,{children:"0"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"3"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"4"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"}"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n \\in \\{0,1,2,3,4\\}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.5782em",verticalAlign:"-0.0391em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mopen",children:"{"}),(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"3"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"4"}),(0,n.jsx)(e.span,{className:"mclose",children:"}"})]})]})]})," i najprawdopodobniej nie ma ich wiÄcej."]}),"\n",(0,n.jsxs)(e.li,{children:["Liczby pierwsze Protha â liczba musi pochodziÄ ze wzoru ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mi,{children:"h"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"k"})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n = h \\cdot 2^k + 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"h"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.9324em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8491em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),", gdzie zar\xf3wno ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"h"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"h"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"h"})]})})]}),", jak i ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"k"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"k"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03148em"},children:"k"})]})})]})," sÄ
liczbami naturalnymi, ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"h"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"h"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"h"})]})})]})," jest nieparzyste oraz ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"k"})]}),(0,n.jsx)(e.mo,{children:">"}),(0,n.jsx)(e.mi,{children:"h"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^k > h"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8882em",verticalAlign:"-0.0391em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8491em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:">"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"h"})]})]})]}),". NajwiÄkszÄ
znanÄ
liczbÄ
pierwszÄ
Protha (luty 2022) jest ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"11"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"9381365"})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"11\\cdot2^{9381365}+1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"11"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"9381365"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),"."]}),"\n",(0,n.jsxs)(e.li,{children:["Factorial prime (nie znam polskiej nazwy; factorial to silnia po angielsku) â sÄ
to liczby pierwsze o 1 wiÄksze lub mniejsze od silnii z dowolnej liczby (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"!"}),(0,n.jsx)(e.mo,{children:"\xb1"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n! \\pm 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mclose",children:"!"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\xb1"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),"). Silnia (oznaczana symbolem ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mo,{stretchy:"false",children:"!"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"!"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mclose",children:"!"})]})})]}),") to iloczyn kolejnych liczb naturalnych. Aktualnie znamy 27 takich liczb po odjÄciu 1 oraz 23 po dodaniu 1. NajwiÄksza znana to ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"308084"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"!"}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"308084! + 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord",children:"308084"}),(0,n.jsx)(e.span,{className:"mclose",children:"!"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"+"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),"."]}),"\n",(0,n.jsxs)(e.li,{children:["Primorial prime (analogicznie jak wyżej, nie znam dobrego tÅumaczenia) â sÄ
to liczby pierwsze o 1 wiÄksze lub o 1 mniejsze od primorial z ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mi,{children:"n"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p_n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]})]})})]})," (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"#"}),(0,n.jsx)(e.mo,{children:"\xb1"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p_n\\# \\pm 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]}),(0,n.jsx)(e.span,{className:"mord",children:"#"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\xb1"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),"). Primorial (oznaczany symbolem ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{mathvariant:"normal",children:"#"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\#"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord",children:"#"})]})})]}),") to iloczyn kolejnych liczb pierwszych. NajwiÄksza znana to ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"3267113"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"#"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"3267113\\# - 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord",children:"3267113#"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})," (dla ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"234725"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n=234725"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"234725"})]})]})]}),")."]}),"\n"]}),"\n",(0,n.jsx)(e.p,{children:"Zauważ ich wsp\xf3lnÄ
cechÄ (oraz liczb Mersenne'a): zawsze sÄ
o 1 wiÄksze lub mniejsze od innej znanej nam liczby uzyskiwanej z prostego wzoru."}),"\n",(0,n.jsxs)(e.p,{children:["Wyżej wymienione liczby pierwsze, a także inne rodzaje, aktualnie wyszukuje inny projekt rozproszonych obliczeÅ: ",(0,n.jsx)(e.a,{href:"https://www.primegrid.com/",children:"PrimeGrid"}),". Analogicznie jak w GIMPS, każdy może wspom\xf3c wyszukiwanie kolejnych dużych liczb pierwszych mocÄ
swojego komputera."]}),"\n",(0,n.jsxs)(e.h2,{id:"maÅe-twierdzenie-fermata",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#maÅe-twierdzenie-fermata",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"MaÅe twierdzenie Fermata"]}),"\n",(0,n.jsx)(e.p,{children:"Nazwisko Pierre'a de Fermata, XVII-wiecznego francuskiego matematyka-samouka, na pewno już nie raz sÅyszaÅeÅ(-aÅ), nawet pomijajÄ
c wyżej opisane liczby Fermata. Mamy m.in. spiralÄ Fermata, punkt Fermata, zasadÄ Fermata, twierdzenie Fermata o sumie dw\xf3ch kwadrat\xf3w czy wielkie twierdzenie Fermata. Nas jednak, z punktu widzenia liczb pierwszych, interesujÄ
dwa zagadnienia: maÅe twierdzenie Fermata oraz test pierwszoÅci Fermata. Skupmy siÄ na tym pierwszym. Twierdzenie brzmi nastÄpujÄ
co:"}),"\n",(0,n.jsxs)(e.p,{children:["JeÅli ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"p"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})})]})," jest liczbÄ
pierwszÄ
, to dla dowolnej liczby caÅkowitej ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"a"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"})]})})]})," liczba ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsx)(e.mi,{children:"p"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mi,{children:"a"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a^p-a"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7477em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.6644em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"p"})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"})]})]})]})," jest podzielna przez ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"p"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})})]}),"."]}),"\n",(0,n.jsx)(e.p,{children:"Możemy to zapisaÄ w arytmetyce modularnej (czyli z zawijaniem siÄ liczb po osiÄ
gniÄciu liczby zwanej moduÅem; tak jak przy dodawaniu czasu) nastÄpujÄ
co:"}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsx)(e.mi,{children:"p"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mn,{children:"0"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mspace,{width:"0.3333em"}),(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\displaystyle a^{p}-a\\equiv 0{\\pmod {p}}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7977em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.7144em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"p"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4637em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â¡"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mspace allowbreak"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"1em"}}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathrm",children:"mod"})})}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.3333em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})]})]})]})}),"\n",(0,n.jsxs)(e.p,{children:["IdÄ
c dalej, jeÅli ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"a"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"})]})})]})," nie jest podzielne przez ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"p"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"})]})})]}),", maÅe twierdzenie Fermata jest r\xf3wnoważne stwierdzeniu, że ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a^{p-1}-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
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to:"}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]})]}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a^{p-1} \\mod{p} = 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8641em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8641em"}
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liczbami pierwszymi (majÄ
wiÄcej niż 2 dzielniki), ale speÅniajÄ
niekt\xf3re wÅaÅciwoÅci charakterystyczne dla nich."}),"\n",(0,n.jsxs)(e.p,{children:["NajczÄÅciej spotkaÄ siÄ możemy z liczbami pseudopierwszymi Fermata. SÄ
to liczby, kt\xf3re speÅniajÄ
warunki maÅego twierdzenia Fermata, jednak nie sÄ
liczbami pierwszymi. Na przykÅad najmniejsza liczba pseudopierwsza Fermata przy podstawie 2 (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a = 2"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"})]})]})]}),", fachowo oznacza siÄ takie liczby jako 2-PRP) to 341. Nie jest to liczba pierwsza, bo jest podzielna przez 11 i 31, jednak możemy zauważyÄ zgodnie z maÅym twierdzeniem Fermata, że ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"340"})]}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"0.4444em"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mspace,{width:"0.3333em"}),(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^{340}\\equiv 1{\\pmod {p}}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"340"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â¡"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mspace allowbreak"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.4444em"}}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathrm",children:"mod"})})}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.3333em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})]})]})]}),"."]}),"\n",(0,n.jsxs)(e.p,{children:["Liczb pseudopierwszych jest nieskoÅczenie wiele, ale w zależnoÅci od ich kategorii i podstawy nie musi byÄ ich dużo w por\xf3wnaniu do liczb pierwszych. PrzykÅadowo, poniżej ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mo,{separator:"true",children:","}),(0,n.jsx)(e.mn,{children:"5"}),(0,n.jsx)(e.mo,{children:"â
"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"10"}),(0,n.jsx)(e.mn,{children:"10"})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2,5 \\cdot 10^{10}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8389em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord",children:"5"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â
"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"0"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"10"})})})]})})})})})]})]})]})]})," znajdziemy okoÅo miliard liczb pierwszych, podczas gdy zaledwie 21 853 liczby pseudopierwsze Fermata o podstawie 2."]}),"\n",(0,n.jsxs)(e.h2,{id:"testowanie-dużych-liczb-pierwszych",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#testowanie-dużych-liczb-pierwszych",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Testowanie dużych liczb pierwszych"]}),"\n",(0,n.jsx)(e.p,{children:"W przypadku dużych liczb pierwszych testowanie metodÄ
naiwnÄ
, kt\xf3rÄ
pokazaÅem w pierwszym artykule, może byÄ bardzo czasochÅonne. Tymczasem, jak już wspomniaÅem na samym poczÄ
tku, dużo algorytm\xf3w polega na tym, że jesteÅmy w stanie znaleÅºÄ liczby pierwsze szybko, a skoro znaleźÄ, to też przetestowaÄ je."}),"\n",(0,n.jsx)(e.p,{children:"W tym kontekÅcie możemy m\xf3wiÄ o kilku rodzajach test\xf3w pierwszoÅci:"}),"\n",(0,n.jsxs)(e.ul,{children:["\n",(0,n.jsx)(e.li,{children:"(Silne) testy pseudopierwszoÅci"}),"\n",(0,n.jsx)(e.li,{children:"Klasyczne testy"}),"\n",(0,n.jsx)(e.li,{children:"Testy og\xf3lnego przeznaczenia"}),"\n"]}),"\n",(0,n.jsx)(e.p,{children:"W ramach tego artykuÅu kr\xf3tko opowiem o tych rodzajach test\xf3w, a także o tym, jakie testy możemy tutaj wyr\xf3żniÄ i na czym mniej wiÄcej polegajÄ
. Bardziej szczeg\xf3Åowy opis dziaÅania jednego z najpowszechniej stosowanych sposob\xf3w przedstawiÄ
1w nastÄpnym artykule."}),"\n",(0,n.jsxs)(e.h3,{id:"silne-testy-pseudopierwszoÅci",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#silne-testy-pseudopierwszoÅci",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"(Silne) testy pseudopierwszoÅci"]}),"\n",(0,n.jsx)(e.p,{children:"Testy pseudopierwszoÅci nie wskażÄ
nam, że liczba jest na pewno pierwsza, ponieważ w najgorszym wypadku może byÄ to liczba pseudopierwsza. Jednak testy takie sÄ
bardzo szybkie, stÄ
d też czÄsto wykorzystuje siÄ je w praktyce (np. w algorytmie RSA)."}),"\n",(0,n.jsxs)(e.h4,{id:"test-pierwszoÅci-fermata",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#test-pierwszoÅci-fermata",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Test pierwszoÅci Fermata"]}),"\n",(0,n.jsxs)(e.p,{children:["Najbardziej znanym jest tutaj test pierwszoÅci Fermata. Polega na sprawdzaniu losowych liczb (sami okreÅlamy, ile razy sprawdzamy, za pomocÄ
wartoÅci ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"k"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"k"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03148em"},children:"k"})]})})]}),") i naszej prawdopodobnej liczby pierwszej wedÅug maÅego twierdzenia Fermata. Robimy to nastÄpujÄ
co:"]}),"\n",(0,n.jsxs)(e.ul,{children:["\n",(0,n.jsxs)(e.li,{children:["Wybierz losowe ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"a"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"})]})})]})," w zakresie ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mo,{children:"⩽"}),(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsx)(e.mo,{children:"⩽"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2 \\leqslant a \\leqslant n-2"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7811em",verticalAlign:"-0.1367em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel amsrm",children:"⩽"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7733em",verticalAlign:"-0.1367em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel amsrm",children:"⩽"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"2"})]})]})]}),"."]}),"\n",(0,n.jsxs)(e.li,{children:["JeÅli ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]})]}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"0.6667em"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{mathvariant:"normal",children:"â "}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a^{n-1} \\mod{n} \\neq 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"}),(0,n.jsx)(e.span,{className:"mbin mtight",children:"â"}),(0,n.jsx)(e.span,{className:"mord mtight",children:"1"})]})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace allowbreak"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.6667em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathrm",children:"mod"})})}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsxs)(e.span,{className:"mrel",children:[(0,n.jsx)(e.span,{className:"mrel",children:(0,n.jsx)(e.span,{className:"mord vbox",children:(0,n.jsx)(e.span,{className:"thinbox",children:(0,n.jsxs)(e.span,{className:"rlap",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"inner",children:(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mrel",children:"î "})})}),(0,n.jsx)(e.span,{className:"fix"})]})})})}),(0,n.jsx)(e.span,{className:"mrel",children:"="})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),", to ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," jest liczbÄ
zÅożonÄ
i przerwij sprawdzanie."]}),"\n",(0,n.jsxs)(e.li,{children:["Powt\xf3rz powyższe kroki ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"k"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"k"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03148em"},children:"k"})]})})]})," razy. JeÅli warunek nie zostaÅ nigdy speÅniony, ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," jest prawdopodobnie pierwsze."]}),"\n"]}),"\n",(0,n.jsxs)(e.h4,{id:"test-millera-rabina",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#test-millera-rabina",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Test Millera-Rabina"]}),"\n",(0,n.jsx)(e.p,{children:"Niestety, mimo że jak wczeÅniej pisaÅem, liczb pseudopierwszych Fermata nie jest bardzo dużo, to jednak do praktycznych zastosowaÅ jest ich zbyt wiele, aby test byÅ wystarczajÄ
co wiarygodny. Dlatego też powstaÅy tzw. silne testy pseudopierwszoÅci, gdzie prawdopodobieÅstwo uznania liczby zÅożonej za pierwszÄ
jest jeszcze rzadsze. Szeroko stosowanymi standardami w praktycznych zastosowaniach sÄ
testy Baillie-PSW oraz Millera-Rabina. Ten drugi jest szczeg\xf3lnie czÄsto spotykany ze wzglÄdu na jego prostotÄ oraz szybkoÅÄ."}),"\n",(0,n.jsxs)(e.p,{children:["Test Millera-Rabina opiszÄ bardziej szczeg\xf3Åowo w n
1astÄpnym artykule, jednak bardzo pobieżnie m\xf3wiÄ
c, jego idea opiera siÄ na rozbiciu sprawdzanej liczby ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," na ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mi,{children:"s"})]}),(0,n.jsx)(e.mo,{children:"â
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marginesu bÅÄdu. JeÅli chcielibyÅmy opublikowaÄ odkrycie nowej bardzo dużej liczby pierwszej, nie wystarczy przeprowadziÄ test\xf3w pseudopierwszoÅci. Z drugiej strony, metoda naiwna też siÄ nie sprawdzi, bo wymagaÅaby zbyt dużo obliczeÅ. Tutaj pojawia siÄ zagadnienie klasycznych test\xf3w pierwszoÅci."}),"\n",(0,n.jsxs)(e.p,{children:["Idea tych test\xf3w opiera siÄ na tym, że nie sprawdzimy nimi każdej możliwej liczby, czy jest pierwsza. Stosuje siÄ je tylko na wybrane przypadki. Dlatego teraz przypomnij sobie, kiedy wymieniaÅem rodzaje liczb pierwszych, albo też zobacz w Internecie dowolnÄ
tabelÄ najwiÄkszych znanych liczb pierwszych. Zwykle mamy tam zapis ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{children:"\xb1"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pm 1"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord",children:"\xb1"}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})})]})," inna liczba. W takich przypadkach okazuje siÄ, że liczba o 1 mniejsza/wiÄksza od liczby pierwszej jest bardzo prosta do rozbicia na dzielniki. Na tej podstawie, jesteÅmy potem w stanie okreÅliÄ pierwszoÅÄ kolejnej/poprzedniej liczby."]}),"\n",(0,n.jsxs)(e.p,{children:["Testy klasyczne możemy podzieliÄ na testy ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),", czyli te, gdzie sprawdzamy liczbÄ o 1 mniejszÄ
(a wiÄc liczbÄ zapisanÄ
jako ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"p"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mtext,{children:"co"}),(0,n.jsxs)(e.mover,{accent:"true",children:[(0,n.jsx)(e.mtext,{children:"s"}),(0,n.jsx)(e.mo,{children:"Ë"})]})]}),(0,n.jsx)(e.mo,{children:"+"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"p = \\text{coÅ} + 1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7
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Duży udziaÅ w opracowaniu obu rodzaj\xf3w tych test\xf3w miaÅ jeszcze inny francuski matematyk â \xc9douard Lucas, kt\xf3rego możesz kojarzyÄ z wymyÅlenia zagadki ",(0,n.jsx)(e.a,{href:"/post/wieze-hanoi",children:"wieży Hanoi"}),"."]}),"\n",(0,n.jsxs)(e.h4,{id:"n---1",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#n---1",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"n - 1"]}),"\n",(0,n.jsxs)(e.p,{children:["W przypadku ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})," testy te bazujÄ
czÄsto na zmodyfikowanym przez Lucasa maÅym twierdzeniu Fermata, gdzie zakÅadamy, że liczba ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"n"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"})]})})]})," jest pierwsza, jeÅli dla każdego dzielnika (bÄdÄ
cego liczbÄ
pierwszÄ
) ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})," istnieje taka liczba caÅkowita a, kt\xf3ra speÅnia jednoczeÅnie dwa warunki:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mtable,{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em",children:[(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]})]}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mi,{children:"n"})]})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mrow,{}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"1"})]})})})]}),(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,n.jsx)(e.mi,{children:"q"})]})]}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mtext,{children:"â"}),(0,n.jsx)(e.mi,{children:"n"})]})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mrow,{}),(0,n.jsx)(e.mo,{mathvariant:"normal",children:"â "}),(0,n.jsx)(e.mn,{children:"1"})]})})})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\begin{align*}\na^{n-1} \\mod{n} &= 1 \\\\\na^{(n-1)/q} \\mod{n} &\\neq 1\n\\end{align*}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"3.1221em",verticalAlign:"-1.3111em"}}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsxs)(e.span,{className:"mtable",children:[(0,n.jsx)(e.span,{className:"col-align-r",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsxs)(e.span,{className:"vlist",style:{height:"1.8111em"}
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WedÅug niego n-ta liczba Fermata (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"F"}),(0,n.jsx)(e.mi,{children:"n"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"F_n"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"F"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]})]})})]}),") jest liczbÄ
pierwszÄ
tylko wtedy, gdy:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"3"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"F"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,n.jsx)(e.mn,{children:"2"})]})]}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mspace,{width:"0.3333em"}),(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"F"}),(0,n.jsx)(e.mi,{children:"n"})]}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"3^{(F_n-1)/2} \\equiv -1 \\pmod {F_n}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.938em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"3"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.938em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.113em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mopen mtight",children:"("}),(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.13889em"},children:"F"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1645em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.357em",marginLeft:"-0.1389em",marginRight:"0.0714em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.5em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size3 size1 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",children:"n"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.143em"},children:(0,n.jsx)(e.span,{})})})]})})]}),(0,n.jsx)(e.span,{className:"mbin mtight",children:"â"}),(0,n.jsx)(e.span,{className:"mord mtight",children:"1"}),(0,n.jsx)(e.span,{className:"mclose mtight",children:")"}),(0,n.jsx)(e.span,{className:"mord mtight",children:"/2"})]})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"â¡"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.7278em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord",children:"â"}),(0,n.jsx)(e.span,{className:"mord",children:"1"}),(0,n.jsx)(e.span,{className:"mspace allowbreak"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"1em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord",children:(0,n.jsx)(e.span,{className:"mord mathrm",children:"mod"})})}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.3333em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"F"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}
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jest:"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"a"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,n.jsx)(e.mn,{children:"2"})]})]}),(0,n.jsx)(e.mo,{children:"â¡"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"}),(0,n.jsx)(e.mspace,{}),(0,n.jsx)(e.mspace,{width:"1em"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{mathvariant:"normal",children:"m"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"o"}),(0,n.jsx)(e.mi,{mathvariant:"normal",children:"d"})]}),(0,n.jsx)(e.mspace,{width:"0.3333em"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"a^{(n-1)/2} \\equiv -1 \\pmod{n}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.938em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.938em"}
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najbardziej znane liczby pierwsze â liczby pierwsze Mersenne'a. Tutaj najbardziej znanym sposobem jest test Lucasa-Lehmera. PomijajÄ
c caÅÄ
zawiÅÄ
matematykÄ bÄdÄ
cÄ
podstawÄ
tego testu, można go sprowadziÄ do kilku prostych obliczeÅ."}),"\n",(0,n.jsxs)(e.p,{children:["Zacznijmy od definicji ciÄ
gu liczb naturalnych ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsx)(e.mi,{children:"k"})]})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"S_k"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,n.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0576em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})}),(0,n.jsx)(e.span,{className:"vlist-s",children:"â"})]}),(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,n.jsx)(e.span,{})})})]})})]})]})})]}),":"]}),"\n",(0,n.jsx)(e.span,{className:"katex-display",children:(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msub,{children:[(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsx)(e.mi,{children:"k"})]}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mo,{fence:"true",children:"{"}),(0,n.jsxs)(e.mtable,{rowspacing:"0.36em",columnalign:"left left",columnspacing:"1em",children:[(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,n.jsx)(e.mn,{children:"4"})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mtext,{children:",\xa0gdy\xa0"}),(0,n.jsx)(e.mi,{children:"k"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"0"})]})})})]}),(0,n.jsxs)(e.mtr,{children:[(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msubsup,{children:[(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"k"}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.mn,{children:"2"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"2"})]})})}),(0,n.jsx)(e.mtd,{children:(0,n.jsx)(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mtext,{children:",\xa0gdy\xa0"}),(0,n.jsx)(e.mi,{children:"k"}),(0,n.jsx)(e.mo,{mathvariant:"normal",children:"â "}),(0,n.jsx)(e.mn,{children:"0"})]})})})]})]})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"S_{k}={\\begin{cases}4&{\\text{, gdy }}k=0\\\\S_{k-1}^{2}-2&{\\text{, gdy }}k\\neq 0\\end{cases}}"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,n.jsxs)(e.span,{className:"vlist-r",children:[(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"}
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liczbÄ pierwszÄ
Mersenne'a (",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mn,{children:"2"}),(0,n.jsx)(e.mn,{children:"21701"})]}),(0,n.jsx)(e.mo,{children:"â"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"2^{21701}-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8974em",verticalAlign:"-0.0833em"}}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord",children:"2"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:(0,n.jsx)(e.span,{className:"mord mtight",children:"21701"})})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"â"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),") znalazÅa dw\xf3jka licealist\xf3w (Landon Curt Noll i Laura Nickel)."]}),"\n",(0,n.jsxs)(e.h3,{id:"testy-og\xf3lnego-przeznaczenia",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#testy-og\xf3lnego-przeznaczenia",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Testy og\xf3lnego przeznaczenia"]}),"\n",(0,n.jsx)(e.p,{children:"W tej kategorii znajdziemy testy pierwszoÅci, zwykle doÅÄ nowoczesne, kt\xf3re albo stanowiÄ
usprawnione wersje wczeÅniej opisanych, albo
1stosujÄ
zupeÅnie inne podejÅcie do sprawdzania pierwszoÅci z pominiÄciem dzielenia. W przypadku tych sposob\xf3w mamy czÄsto do czynienia z doÅÄ zaawansowanÄ
matematykÄ
, dlatego wymieniÄ je tylko z nazwy:"}),"\n",(0,n.jsxs)(e.ul,{children:["\n",(0,n.jsx)(e.li,{children:"APR oraz APR-CL â sÄ
dalszymi rozwiniÄciami idei test\xf3w klasycznych, stÄ
d czÄsto nazywa siÄ je testami neoklasycznymi."}),"\n",(0,n.jsx)(e.li,{children:"ECPP â do sprawdzenia pierwszoÅci wykorzystuje krzywe eliptyczne."}),"\n",(0,n.jsx)(e.li,{children:"AKS â podobnie jak test Rabina-Millera wykorzystuje zmodyfikowanÄ
wersjÄ maÅego twierdzenia Fermata, jednak tutaj mamy do czynienia z testem deterministycznym. Jego zaletami sÄ
szybkoÅÄ dziaÅania oraz to, że można go w miarÄ Åatwo zaprogramowaÄ."}),"\n"]}),"\n",(0,n.jsxs)(e.h2,{id:"podsumowanie",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#podsumowanie",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Podsumowanie"]}),"\n",(0,n.jsx)(e.p,{children:"Jak mogÅeÅ(-aÅ) zobaczyÄ w tym artykule, w wÄ
tku liczb pierwszych mamy bardzo wiele ciekawych odkryÄ. Nie tylko coraz to wiÄksze liczby, ale także wÅasnoÅci, jakie liczby pierwsze posiadajÄ
. Mimo że wciÄ
ż nie znamy wzoru na liczbÄ pierwszÄ
, odkrycia sprawiÅy, że szukanie coraz to wiÄkszych liczb jest prostsze i szybsze."}),"\n",(0,n.jsx)(e.p,{children:"A po co w og\xf3le szukaÄ tak dużych liczb pierwszych, pomijajÄ
c zastosowanie w kryptografii, o kt\xf3rym wspominaÅem wiele razy? C\xf3ż, jest to już matematyczna tradycja, kt\xf3rÄ
zajmowaÅo siÄ wielu znanych matematyk\xf3w wspominanych przeze mnie nie raz w tym artykule. Jednak nie tylko: w historii szukanie dużych liczb pierwszych miaÅo skutki uboczne w postaci nowych twierdzeÅ czy algorytm\xf3w. A jak to do Ciebie nie przemawia, to dla z
1nalazc\xf3w rekordowych liczb pierwszych sÄ
przewidziane nagrody pieniÄżne â 100 000 $ za pierwszÄ
odkrytÄ
liczbÄ z dziesiÄcioma milionami cyfr, 150 000 $ za pierwszÄ
z miliardem cyfr, 250 000 $ za pierwszÄ
z bilionem cyfr."}),"\n",(0,n.jsxs)(e.h2,{id:"literatura",children:[(0,n.jsx)(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#literatura",children:(0,n.jsx)(e.span,{className:"icon icon-link"})}),"Literatura"]}),"\n",(0,n.jsxs)(e.ul,{children:["\n",(0,n.jsxs)(e.li,{children:["Mollin, R. A. (2002). ",(0,n.jsx)(e.em,{children:"A Brief History of Factoring and Primality Testing B. C. (Before Computers)"}),". Mathematics Magazine, 75(1), 18. ",(0,n.jsx)(e.a,{href:"https://doi.org/10.2307%2F3219180",children:"doi:10.2307/3219180"})]}),"\n",(0,n.jsxs)(e.li,{children:["Granville, A.; Martin, G. (2006). âPrime Number Racesâ. ",(0,n.jsx)(e.em,{children:"American Mathematical Monthly"}),". 113 (1): 1â33. ",(0,n.jsx)(e.a,{href:"https://doi.org/10.2307%2F27641834",children:"doi:10.2307/27641834"}),"."]}),"\n",(0,n.jsxs)(e.li,{children:["Knuth, D. E. âFactoring into Primesâ w ",(0,n.jsx)(e.em,{children:"The art of computer programming: Volume 2."}),". Addison-Wesley, 2011, s. 380-418"]}),"\n",(0,n.jsxs)(e.li,{children:["Cormen, T. H.; Leiserson, C. E.; Rivest R. L.; Stein C. âPrimality Testingâ w ",(0,n.jsx)(e.em,{children:"Introduction to algorithms, 3rd ed."}),". MIT Press, Cambridge, MA, U.S.A., s. 965-975."]}),"\n",(0,n.jsxs)(e.li,{children:["Weisstein, Eric W. âPrime Counting Function.â From *MathWorld--*A Wolfram Web Resource. ",(0,n.jsx)(e.a,{href:"https://mathworld.wolfram.com/PrimeCountingFunction.html",children:"https://mathworld.wolfram.com/PrimeCountingFunction.html"})]}),"\n",(0,n.jsxs)(e.li,{children:["Caldwell C., ",(0,n.jsx)(e.em,{children:"Finding primes & proving primality â How to Prove Primality"}),", ",(0,n.jsx)(e.a,{href:"https://primes.utm.edu/prove/merged.html",children:"https://primes.utm.edu/prove/merged.html"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n",(0,n.jsxs)(e.li,{children:["Mersenne Primes: History, Theorems and Lists, ",(0,n.jsx)(e.a,{href:"https://primes.utm.edu/mersenne/index.html",children:"https://primes.utm.edu/mersenne/index.html"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n",(0,n.jsxs)(e.li,{children:["Mathematics and Research Strategy, ",(0,n.jsx)(e.a,{href:"https://www.mersenne.org/various/math.php",children:"https://www.mersenne.org/various/math.php"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n",(0,n.jsxs)(e.li,{children:["Ulam spiral, ",(0,n.jsx)(e.a,{href:"https://en.wikipedia.org/w/index.php?title=Ulam_spiral&oldid=1038024272",children:"https://en.wikipedia.org/w/index.php?title=Ulam_spiral&oldid=1038024272"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n",(0,n.jsxs)(e.li,{children:["Fermat's little theorem, ",(0,n.jsx)(e.a,{href:"https://en.wikipedia.org/w/index.php?title=Fermat%27s_little_theorem&oldid=1054384646",children:"https://en.wikipedia.org/w/index.php?title=Fermat%27s_little_theorem&oldid=1054384646"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n",(0,n.jsxs)(e.li,{children:["MillerâRabin primality test, ",(0,n.jsx)(e.a,{href:"https://en.wikipedia.org/w/index.php?title=Miller%E2%80%93Rabin_primality_test&oldid=1058489090",children:"https://en.wikipedia.org/w/index.php?title=Miller%E2%80%93Rabin_primality_test&oldid=1058489090"})," (ostatnie odwiedziny 12.02.2022)."]}),"\n"]})]})}let N={Component:function(){let s=arguments.length>0&&void 0!==arguments[0]?arguments[0]:{},{wrapper:e}={...(0,i.R)(),...s.components};return e?(0,n.jsx)(e,{...s,children:(0,n.jsx)(o,{...s})}):o(s)},meta:{title:"Duże liczby pierwsze",abstract:"SkÄ
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tku serii"}),", w kryptografii wykorzystuje siÄ liczby pierwsze 2048-bitowe, wiÄc w systemie dziesiÄtnym mogÄ
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