A197635 Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

2, 3, 14771

Offset

1,1

Comments

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.

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

Links

Table of n, a(n) for n=1..3.

Carlos Rivera, Problem 59. Wieferich-non-Wilson primes, The Prime Puzzles and Problems Connection.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Formula

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

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

Example

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

Crossrefs

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

Sequence in context: A146026 A115640 A212494 * A171161 A101445 A128668

Adjacent sequences: A197632 A197633 A197634 * A197636 A197637 A197638

Keyword

nonn,hard,more,bref

Author

Jonathan Sondow, Oct 16 2011

Status

approved