A197635 - OEIS

%I #31 Nov 08 2024 07:54:49

%S 2,3,14771

%N Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

%C A Wieferich prime base a is a prime p satisfying a^(p-1) == 1 (mod p^2). A non-Wilson prime p is called a Wieferich-non-Wilson prime if p is a Wieferich prime base w_p, where w_p = ((p-1)!+1)/p is the Wilson quotient of p.

%C Michael Mossinghoff has computed that if a 4th Wieferich-non-Wilson prime exists, it is > 10^7.

%H Carlos Rivera, <a href="https://www.primepuzzles.net/problems/prob_059.htm">Problem 59. Wieferich-non-Wilson primes</a>, The Prime Puzzles and Problems Connection.

%H Jonathan Sondow, <a href="http://arxiv.org/abs/1110.3113">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113

%H Jonathan Sondow, <a href="http://link.springer.com/chapter/10.1007%2F978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

%F A197634(A197637(a(n))) = 0.

%F (((p-1)!+1)/p)^(p-1) == 1 (mod p^2), where p = a(n).

%e The first two non-Wilson primes are 2 and 3, whose Fermat-Wilson quotients are 0. Since 2 and 3 divide 0, they are members.

%e The 1728th non-Wilson prime is prime(1731) = 14771, and A197634(1728) = 0, so 14771 is a member.

%Y Cf. A007540, A197633, A197634, A197636, A197637.

%K nonn,hard,more,bref

%O 1,1

%A _Jonathan Sondow_, Oct 16 2011