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

Notes

  1. ↑ Conjecture 4. Fermat primes are finite - Pepin tests story, according to Leonid Durman
  2. ↑ Wilfrid Keller: Fermat factoring status
  3. ↑ R. M. Robinson (1954): Mersenne and Fermat numbers
  4. ↑ Richard E. Crandall, Ernst W. Mayer & Jason S. Papadopoulos, The twenty-fourth Fermat number is composite (2003)

References

External links