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| In [[mathematics]], '''Pépin's test''' is a [[primality test]], which can be used to determine whether a [[Fermat number]] is [[prime number|prime]]. It is a variant of [[Proth's theorem|Proth's test]]. The test is named for a French mathematician, [[Théophile Pépin]].
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| ==Description of the test==
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| Let <math>F_n=2^{2^n}+1</math> be the ''n''th Fermat number. Pépin's test states that for ''n'' > 0,
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| :<math>F_n</math> is prime if and only if <math>3^{(F_n-1)/2}\equiv-1\pmod{F_n}.</math>
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| The expression <math>3^{(F_n-1)/2}</math> can be evaluated modulo <math>F_n</math> by [[exponentiation by squaring|repeated squaring]]. This makes the test a fast [[polynomial-time]] algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat numbers can be tested in a reasonable amount of time and space.
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| Other bases may be used in place of 3, for example 5, 6, 7, or 10 {{OEIS|id=A129802}}.
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| ==Proof of correctness==
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| Sufficiency: assume that the congruence
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| :<math>3^{(F_n-1)/2}\equiv-1\pmod{F_n}</math>
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| holds. Then <math>3^{F_n-1}\equiv1\pmod{F_n}</math>, thus the [[multiplicative order]] of 3 modulo <math>F_n</math> divides <math>F_n-1=2^{2^n}</math>, which is a power of two. On the other hand, the order does not divide <math>(F_n-1)/2</math>, and therefore it must be equal to <math>F_n-1</math>. In particular, there are at least <math>F_n-1</math> numbers below <math>F_n</math> coprime to <math>F_n</math>, and this can happen only if <math>F_n</math> is prime.
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| Necessity: assume that <math>F_n</math> is prime. By [[Euler's criterion]],
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| :<math>3^{(F_n-1)/2}\equiv\left(\frac3{F_n}\right)\pmod{F_n}</math>,
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| where <math>\left(\frac3{F_n}\right)</math> is the [[Legendre symbol]]. By repeated squaring, we find that <math>2^{2^n}\equiv1\pmod3</math>, thus <math>F_n\equiv2\pmod3</math>, and <math>\left(\frac{F_n}3\right)=-1</math>.
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| As <math>F_n\equiv1\pmod4</math>, we conclude <math>\left(\frac3{F_n}\right)=-1</math> from the [[law of quadratic reciprocity]].
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| == Historical Pépin tests ==
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| Because of the sparsity of the Fermat numbers, the Pépin test has only been run seven times (on Fermat numbers whose primality statuses were not already known).<ref>[http://www.primepuzzles.net/conjectures/conj_004.htm Conjecture 4. Fermat primes are finite - Pepin tests story, according to Leonid Durman]</ref><ref>Wilfrid Keller: [http://www.prothsearch.net/fermat.html Fermat factoring status]</ref>
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| Mayer, Papadopoulos and Crandall speculate that in fact, because of the size of the still undetermined Fermat numbers, it will take decades before technology allows any more Pépin tests to be run.<ref>Richard E. Crandall, Ernst W. Mayer & Jason S. Papadopoulos, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01479-5/S0025-5718-02-01479-5.pdf The twenty-fourth Fermat number is composite (2003)]</ref> {{As of|2012}} the smallest untested Fermat number with no known prime factor is <math>F_{33}</math> which has 2,585,827,973 digits.
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| {| class="wikitable" border="1"
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| |-
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| ! Year
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| ! Provers
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| ! Fermat number
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| ! Pépin test result
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| ! Factor found later?
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| |-
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| | 1905
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| | [[James C. Morehead|Morehead]] & [[Alfred Western|Western]]
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| | <math>F_{7}</math>
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| | composite
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| | Yes (1970)
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| |-
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| | 1909
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| | Morehead & Western
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| | <math>F_{8}</math>
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| | composite
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| | Yes (1980)
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| |-
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| | 1960
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| | [[G. A. Paxson|Paxson]]
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| | <math>F_{13}</math>
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| | composite
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| | Yes (1974)
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| |-
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| | 1961
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| | [[John Selfridge|Selfridge]] & [[Alexander Hurwitz|Hurwitz]]
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| | <math>F_{14}</math>
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| | composite
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| | Yes (2010)
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| |-
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| | 1987
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| | [[Duncan Buell|Buell]] & [[Jeffrey Young|Young]]
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| | <math>F_{20}</math>
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| | composite
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| | No
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| |-
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| | 1993
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| | [[Richard Crandall|Crandall]], Doenias, Norrie & Young
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| | <math>F_{22}</math>
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| | composite
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| | Yes (2010)
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| |-
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| | 1999
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| | Mayer, Papadopoulos & Crandall
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| | <math>F_{24}</math>
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| | composite
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| | No
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| |-
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| |}
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * P. Pépin, ''Sur la formule <math>2^{2^n}+1</math>'', ''Comptes Rendus Acad. Sci. Paris'' 85 (1877), pp. 329–333.
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| == External links ==
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| *[http://primes.utm.edu/glossary/page.php?sort=PepinsTest The Prime Glossary: Pepin's test]
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| {{number theoretic algorithms}}
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| {{DEFAULTSORT:Pepin's Test}}
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| [[Category:Primality tests]]
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