A152156 - OEIS

%I #6 Apr 03 2023 10:36:11

%S -1,0,-1,-1,-1,-810129131,-1220845804166146754,

%T 6964187975677595099156927503004398881,

%U 14553806122642016769237504145596730952769427034161327480375008633175279343120

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

%C 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)).

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

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

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

%H Dennis Martin, <a href="/A152156/b152156.txt">Table of n, a(n) for n = 0..11</a>

%H Chris Caldwell, The Prime Pages: <a href="https://t5k.org/glossary/page.php?sort=PepinsTest">Pepin's Test</a>.

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

%e a(4) = 5^(32768) (mod 65537) = 65536 = -1 (mod F(4)), therefore F(4) is prime.

%e a(5) = 5^(2147483648) (mod 4294967297) = -810129131 (mod F(5)), therefore F(5) is composite.

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

%K sign

%O 0,6

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