Tekst źródłowy strony Symbol Legendre’a

'''Symbol Legendre’a''' – [[Funkcja multiplikatywna|funkcja ściśle multiplikatywna]] stosowana w [[Teoria liczb|teorii liczb]], oznaczana <math>(a|p)</math> lub <math>\left( \frac{a}{p} \right)</math><ref name=":0">{{Cytuj |autor = Adam Neugebauer |tytuł = Matematyka olimpijska. 1, Algebra i teoria liczb |data = 2018 |data dostępu = 2022-08-14 |isbn = 978-83-7267-710-5 |wydanie = 2 |miejsce = Kraków |wydawca = Wydawnictwo Szkolne OMEGA |s = 204–211 |oclc = 1055646686}}</ref><ref name=":1">{{Cytuj |autor = Władysław Narkiewicz |tytuł = Teoria liczb |data = 2003 |data dostępu = 2022-08-14 |isbn = 83-01-14015-1 |wydanie = 3 |miejsce = Warszawa |wydawca = Wydawnictwo Naukowe PWN |s = 61, 63–65, 139, 169 |oclc = 749285993}}</ref><ref name=":2">{{Cytuj |autor = Eric W. Weisstein |tytuł = Legendre Symbol |data dostępu = 2022-08-14 |opublikowany = mathworld.wolfram.com |url = https://mathworld.wolfram.com/ |język = en}}</ref>.

Wprowadzony w 1798 przez [[Adrien-Marie Legendre|Legendre’a]]<ref>{{Cytuj |autor = A. M. (Adrien Marie) Legendre |tytuł = Essai sur la théorie des nombres |data = 1798 |data dostępu = 2022-08-14 |wydawca = Paris, Duprat |url = http://archive.org/details/essaisurlathor00lege}}</ref>. Jego uogólnieniem jest [[symbol Jacobiego]].

== Definicja ==
Niech <math>p>2</math> będzie [[Liczby pierwsze|liczbą pierwszą]]. Liczbę <math>a</math> niebędącą [[Wielokrotność|wielokrotnością]] <math>p</math> nazwiemy [[Reszta kwadratowa modulo|resztą kwadratową modulo]] <math>p,</math> jeśli istnieje [[Liczby całkowite|liczba całkowita]] <math>t</math> taka, że <math>a \equiv t^2 \pmod p.</math> Fakt ten oznaczymy <math>a\mathrm{R}p.</math> Jeśli taka liczba <math>t</math> nie istnieje, liczbę <math>a</math> nazywamy [[Niereszta kwadratowa modulo|nieresztą kwadratową modulo]] <math>p</math><ref name=":0" />, w artykule oznaczamy to jako <math>a\mathrm{N}p.</math> Wielokrotności liczby <math>p</math> nie zaliczamy ani do reszt ani do niereszt<ref name=":1" />.

<math>\left(\frac{a}{p}\right) =
\begin{cases}
 1 & \text{gdy } a\mathrm{R}p, \\
-1 & \text{gdy } a\mathrm{N}p, \\
 0 & \text{gdy } a \equiv 0 \pmod p.
\end{cases}</math>

Czasami za dziedzinę funkcji nie przyjmuje się wielokrotności <math>p</math><ref name=":0" /><ref name=":1" /><ref name=":2" />.

== Własności ==
* Jeśli <math>a \equiv b \pmod p,</math> to <math>\left( \frac a p \right) = \left( \frac b p \right)</math><ref name=":1" />.
* [[Kryterium Eulera]] jest użyteczne do obliczania wartości symbolu oraz jest używane do dowodzenia innych własności:<br><math>
\left( \frac a p \right) \equiv a^{\frac {p-1} 2} \pmod p</math><ref name=":0" /><ref name=":1" /><ref name=":2" />.
* Symbol Legendre’a jest [[Funkcja multiplikatywna|funkcją ściśle multiplikatywną]] licznika: <math>\left( \frac{ab}{p} \right) = \left( \frac a p \right) \left( \frac b p \right).</math> Ta własność jest wnioskiem z kryterium Eulera<ref name=":0" /><ref name=":1" /><ref name=":2" />.
* Najważniejszą własnością jest [[prawo wzajemności reszt kwadratowych]], zwane czasami ''theorema fundamentale'' (twierdzenie podstawowe) lub ''theorema aurerum'' (twierdzenie złote)<ref name=":0" /><ref name=":1" /><ref>{{Cytuj |autor = Eric W. Weisstein |tytuł = Quadratic Reciprocity Theorem |data dostępu = 2022-08-14 |opublikowany = mathworld.wolfram.com |url = https://mathworld.wolfram.com/ |język = en}}</ref>:<br><math>
p \neq q \Rightarrow \left( \frac q p \right) \left( \frac p q \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>
* Tę własność, będącą wnioskiem z kryterium Eulera, nazywa się I uzupełnieniem prawa wzajemności<ref name=":0" /><ref name=":1" />:<br><math>
\left(\frac{-1}{p}\right) =
\begin{cases}
 1 & \text{gdy } p \equiv 1 \pmod 4, \\
-1 & \text{gdy } p \equiv 3 \pmod 4.
\end{cases}</math>
* Istnieje również II uzupełnienie prawa wzajemności<ref name=":0" />:<br><math>
\left(\frac{2}{p}\right) =
\begin{cases}
 1 & \text{gdy } p \equiv \plusmn 1 \pmod 8, \\
-1 & \text{gdy } p \equiv \plusmn 3 \pmod 8.
\end{cases}</math>

== Tabela wartości ==
Tabela przedstawia wartości funkcji dla <math>a \leqslant 30</math> i <math>3 \leqslant p \leqslant 127</math><ref>{{Cytuj |tytuł = Legendre Symbol(LS) Calculator |data dostępu = 2022-08-14 |opublikowany = www.mymathtables.com |url = https://www.mymathtables.com/numbers/legendre-symbol-generator.html}}</ref>.

{| class="wikitable" style="text-align:right;"
!{{diagonal split header|''p''|''a''}}
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|-
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|style="background:#ffccff"| {{0|−}}1
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|style="background:#ffffcc"| {{0|−}}0
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|-
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|style="background:#ffccff"| 1
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|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|-
!67
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
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|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ccffff"| −1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|-
!71
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|-
!73
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|-
!79
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
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|-
!83
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|-
!89
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
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|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
|style="background:#ccffff"| −1
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|style="background:#ffccff"| 1
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|-
!97
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|-
!101
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|style="background:#ffccff"| 1
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|style="background:#ffccff"| 1
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|-
!103
|style="background:#ffccff"| 1
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|-
!107
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!109
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!113
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|style="background:#ffccff"| 1
|}

== Przypisy ==
{{Przypisy}}

== Linki zewnętrzne ==
* [http://math.fau.edu/richman/jacobi.htm Kalkulator symbolu Legendre’a]
* {{otwarty dostęp}} ''[https://encyclopediaofmath.org/wiki/Legendre_symbol Legendre symbol]'' {{lang|en}}, Encyclopedia of Mathematics, encyclopediaofmath.org [dostęp 2024-02-02].

{{Teoria liczb}}

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