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| In [[number theory]], '''Proth's theorem''' is a [[primality test]] for [[Proth number]]s.
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| It states that if ''p'' is a '''Proth number''', of the form ''k''2<sup>''n''</sup> + 1 with ''k'' odd and ''k'' < 2<sup>''n''</sup>, then if for some [[integer]] ''a'',
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| :<math>a^{(p-1)/2}\equiv -1 \mod{p}\,\!</math>
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| then ''p'' is [[prime number|prime]] (called a '''Proth prime'''). This is a practical test because if ''p'' is prime, any chosen ''a'' has about a 50 percent chance of working.
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| If ''a'' is a [[quadratic nonresidue]] modulo ''p'' then the converse is also true, and the test is conclusive. Such an ''a'' may be found by iterating ''a'' over small primes and computing the [[Jacobi symbol]] until:
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| :: <math>\left(\frac{a}{p}\right)=-1</math>
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| ==Numerical examples==
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| Examples of the theorem include:
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| * for ''p'' = 3, 2<sup>1</sup> + 1 = 3 is divisible by 3, so 3 is prime.
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| * for ''p'' = 5, 3<sup>2</sup> + 1 = 10 is divisible by 5, so 5 is prime.
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| * for ''p'' = 13, 5<sup>6</sup> + 1 = 15626 is divisible by 13, so 13 is prime.
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| * for ''p'' = 9, which is not prime, there is no ''a'' such that ''a''<sup>4</sup> + 1 is divisible by 9.
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| The first Proth primes are {{OEIS|id=A080076}}:
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| :3, 5, 13, 17, [[41 (number)|41]], [[97 (number)|97]], [[113 (number)|113]], [[193 (number)|193]], 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153 ….
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| {{As of|2009}}, the largest known Proth prime is 19249 · 2<sup>13018586</sup> + 1, found by [[Seventeen or Bust]]. It has 3,918,990 digits and is the largest known prime which is not a [[Mersenne prime]]. <ref>http://primes.utm.edu/top20/page.php?id=3</ref>
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| ==History==
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| [[François Proth]] (1852–1879) published the theorem around 1878.
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| ==See also==
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| *[[Sierpinski number]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|urlname=ProthsTheorem|title=Proth's Theorem}}
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| {{number theoretic algorithms}}
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| [[Category:Primality tests]]
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| [[Category:Theorems about prime numbers]]
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| [[de:Prothsche Primzahl]]
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| [[nl:Prothgetal]]
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