Pépin's test

In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin.

Description of the test

Let $F_{n}=2^{2^{n}}+1$ be the nth Fermat number. Pépin's test states that for n > 0,

$F_{n}$ is prime if and only if $3^{\frac {F_{n}-1}{2}}\equiv -1{\pmod {F_{n}}}.$

The expression $3^{(F_{n}-1)/2}$ can be evaluated modulo $F_{n}$ by 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, 10, 12, 14, 20, 24, 27, 28, 39, 40, 41, 45, 48 (sequence A129802 in the OEIS).

Proof of correctness

Sufficiency: assume that the congruence

$3^{(F_{n}-1)/2}\equiv -1{\pmod {F_{n}}}$

holds. Then $3^{F_{n}-1}\equiv 1{\pmod {F_{n}}}$, thus the multiplicative order of 3 modulo $F_{n}$ divides $F_{n}-1=2^{2^{n}}$, which is a power of two. On the other hand, the order does not divide $(F_{n}-1)/2$, and therefore it must be equal to $F_{n}-1$. In particular, there are at least $F_{n}-1$ numbers below $F_{n}$ coprime to $F_{n}$, and this can happen only if $F_{n}$ is prime.

Necessity: assume that $F_{n}$ is prime. By Euler's criterion,

$3^{(F_{n}-1)/2}\equiv \left({\frac {3}{F_{n}}}\right){\pmod {F_{n}}}$,

where $\left({\frac {3}{F_{n}}}\right)$ is the Legendre symbol. By repeated squaring, we find that $2^{2^{n}}\equiv 1{\pmod {3}}$, thus $F_{n}\equiv 2{\pmod {3}}$, and $\left({\frac {F_{n}}{3}}\right)=-1$. As $F_{n}\equiv 1{\pmod {4}}$, we conclude $\left({\frac {3}{F_{n}}}\right)=-1$ from the law of quadratic reciprocity.

Historical Pépin tests

Because of the sparsity of the Fermat numbers, the Pépin test has only been run eight times (on Fermat numbers whose primality statuses were not already known). 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. As of 2016 the smallest untested Fermat number with no known prime factor is $F_{33}$ which has 2,585,827,973 digits.

Year	Provers	Fermat number	Pépin test result	Factor found later?
1905	Morehead & Western	$F_{7}$	composite	Yes (1970)
1909	Morehead & Western	$F_{8}$	composite	Yes (1980)
1952	Robinson	$F_{10}$	composite	Yes (1953)
1960	Paxson	$F_{13}$	composite	Yes (1974)
1961	Selfridge & Hurwitz	$F_{14}$	composite	Yes (2010)
1987	Buell & Young	$F_{20}$	composite	No
1993	Crandall, Doenias, Norrie & Young	$F_{22}$	composite	Yes (2010)
1999	Mayer, Papadopoulos & Crandall	$F_{24}$	composite	No

References

• P. Pépin, Sur la formule $2^{2^{n}}+1$, Comptes Rendus Acad. Sci. Paris 85 (1877), pp. 329–333.

External links

• The Prime Glossary: Pepin's test