# 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 $3^{(F_n-1)/2}\equiv-1\pmod{F_n}.$

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, or 10 (sequence A129802 in OEIS).

## Proof of correctness

Sufficiency: assume that the congruence

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

holds. Then $3^{F_n-1}\equiv1\pmod{F_n}$, 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,

$3^{(F_n-1)/2}\equiv\left(\frac3{F_n}\right)\pmod{F_n}$,

where $\left(\frac3{F_n}\right)$ is the Legendre symbol. By repeated squaring, we find that $2^{2^n}\equiv1\pmod3$, thus $F_n\equiv2\pmod3$, and $\left(\frac{F_n}3\right)=-1$. As $F_n\equiv1\pmod4$, we conclude $\left(\frac3{F_n}\right)=-1$ 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).[1][2][3] 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.[4] As of 2012 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.