A152156 Minimal residues of Pepin's Test for Fermat Numbers using either 5 or 10 for the base.

-1, 0, -1, -1, -1, -810129131, -1220845804166146754, 6964187975677595099156927503004398881, 14553806122642016769237504145596730952769427034161327480375008633175279343120

Offset: 0

Author

Dennis Martin (dennis.martin(AT)dptechnology.com), Nov 27 2008

Keywords

• sign

Comments

For n=0 or n>=2 the Fermat Number F(n) is prime if and only if 5^((F(n) - 1)/2) is congruent to -1 (mod F(n)).

5 was the base originally used by Pepin. The base 10 gives the same results.

Any positive integer k for which the Jacobi symbol (k|F(n)) is -1 can be used as the base instead.

Examples

			a(4) = 5^(32768) (mod 65537) = 65536 = -1 (mod F(4)), therefore F(4) is prime.
a(5) = 5^(2147483648) (mod 4294967297) = -810129131 (mod F(5)), therefore F(5) is composite.
		

References

• M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001, pp. 42-43.

Links

• Dennis Martin, Table of n, a(n) for n = 0..11
• Chris Caldwell, The Prime Pages: Pepin's Test.

Crossrefs

Cf. A000215, A019434, A152153, A152154, A152155.

Formula

a(n) = 5^((F(n) - 1)/2) (mod F(n)), where F(n) is the n-th Fermat Number