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en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

2015-01-05T13:56:59Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, and/or concising wikilinks, etc.), & ref cleanup using AWB

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en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9936)

2014-02-12T07:56:08Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9936)

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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]].~~		Alyson Meagher is the name her parents gave her but she doesn't like when people use her complete title. Some time ago she selected to live in Alaska and her mothers and fathers reside nearby. Office supervising is what she does for a residing. Playing badminton is a factor that he is totally addicted to.<br><br>Have a look at my blog ... good psychic - [http://jplusfn.gaplus.kr/xe/qna/78647 visit the up coming internet page],

	~~==Description of the test==~~
	~~Let <math>F_n=2^{2^n}+1</math> be~~ the '~~'n''th Fermat number~~. ~~Pépin's test states that for ''n'' > 0,~~
	~~:<math>F_n</math> is prime if and only if <math>3^{(F_n-1)/2}\equiv-1\pmod{F_n}.</math>~~
	~~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~~.

	~~Other bases may be used in place of 3,~~ for ~~example 5, 6, 7, or 10 {{OEIS\|id=A129802}}~~.

	~~==Proof of correctness==~~

	~~Sufficiency: assume that the congruence~~
	~~:<math>3^{(F_n-1)/2}\equiv-1\pmod{F_n}</math>~~
	~~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.

	~~Necessity: assume~~ that ~~<math>F_n</math>~~ is ~~prime~~. ~~By [[Euler's criterion]],~~
	:<~~math~~>~~3^{(F_n-1)/2}\equiv\left(\frac3{F_n}\right)\pmod{F_n}~~<~~/math~~>,
	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>.
	~~As <math>F_n\equiv1\pmod4</math>, we conclude <math>\left(\frac3{F_n}\right)=-1</math> from the [[law of quadratic reciprocity]]~~.

	~~== Historical Pépin tests ==~~
	~~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>~~
	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.~~

	~~{\| class="wikitable" border="1"~~
	\|-
	~~! Year~~
	~~! Provers~~
	~~! Fermat number~~
	~~! Pépin test result~~
	~~! Factor found later?~~
	\|-
	~~\| 1905~~
	~~\| [[James C. Morehead\|Morehead]] & [[Alfred Western\|Western]]~~
	~~\| <math>F_{7}</math>~~
	~~\| composite~~
	~~\| Yes (1970)~~
	\|-
	~~\| 1909~~
	~~\| Morehead & Western~~
	~~\| <math>F_{8}</math>~~
	~~\| composite~~
	~~\| Yes (1980)~~
	\|-
	~~\| 1960~~
	~~\| [[G. A. Paxson\|Paxson]]~~
	~~\| <math>F_{13}</math>~~
	~~\| composite~~
	~~\| Yes (1974)~~
	\|-
	~~\| 1961~~
	~~\| [[John Selfridge\|Selfridge]] & [[Alexander Hurwitz\|Hurwitz]]~~
	~~\| <math>F_{14}</math>~~
	~~\| composite~~
	~~\| Yes (2010)~~
	\|-
	~~\| 1987~~
	~~\| [[Duncan Buell\|Buell]] & [[Jeffrey Young\|Young]]~~
	~~\| <math>F_{20}</math>~~
	~~\| composite~~
	~~\| No~~
	\|-
	~~\| 1993~~
	~~\| [[Richard Crandall\|Crandall]~~], ~~Doenias, Norrie & Young~~
	~~\| <math>F_{22}</math>~~
	~~\| composite~~
	~~\| Yes (2010)~~
	\|-
	~~\| 1999~~
	~~\| Mayer, Papadopoulos & Crandall~~
	~~\| <math>F_{24}</math>~~
	~~\| composite~~
	~~\| No~~
	\|-
	\|}

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* P. Pépin, ''Sur la formule <math>2^{2^n}+1</math>'', ''Comptes Rendus Acad. Sci. Paris'' 85 (1877), pp. 329–333.

	~~== External links ==~~
	*[http://primes.utm.edu/glossary/page.php?sort=PepinsTest The Prime Glossary: Pepin's test]

	~~{{number theoretic algorithms}}~~

	~~{{DEFAULTSORT:Pepin's Test}}~~
	~~[[Category:Primality tests]]~~

en>Gaba p: /* Further reading */ add link

2013-11-27T12:52:20Z

en>AllCluesKey: Clarified and expanded introduction

2012-08-18T22:58:31Z

Clarified and expanded introduction

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