Here are some problems on "Wieferich-non-Wilson
primes", which I define by combining the notions of Wilson
quotient and Fermat quotient. First, I recall those definitions.

1. By Wilson's Theorem, the
"Wilson quotient" of a prime p, namely,

w_p := ((p-1)! + 1)/p,

is an integer. If w_p is divisible by p, then p is called a
"Wilson prime". The known Wilson primes are 5,13, and 563.

2. By Fermat's Little Theorem, if
a prime p does not divide an integer a, then the "Fermat
quotient", namely,

q_p(a) := (a^{p-1} - 1)/p,

is an integer. If q_p(a) is
divisible by p, then p is called a "Wieferich prime base a". For
example, the known Wieferich primes base 2 are 1093 and 3511.

3. Suppose that the prime p is NOT
a Wilson prime, and that p is a Wieferich prime base a, where a
= w_p. Then I call p a "Wieferich-non-Wilson prime". There are
three up to 10^7, namely, 2, 3, and 14771, according to
computations by Michael Mossinghoff.

**
Problem 1.** Without using a computer, can you show
that 14771 is a Wieferich-non-Wilson prime, i.e., that if p
= 14771, then p divides the "Fermat-Wilson quotient" q_p(w_p)?
(When p = 14771, the number q_p(w_p) has over 800 million
digits.)

**Problem
2. **Is there a Wieferich-non-Wilson prime greater than
10^7? Are there infinitely many?

**Problem
3.** Can you prove that infinitely many primes are NOT Wieferich-non-Wilson
primes? Can you prove it assuming the ABC conjecture? (Silverman
has proved that the ABC conjecture implies that infinitely many
primes are not Wieferich primes base 2.)

__________

Note: The third sections of my
slides at https://db.tt/NslQNFrS and
my paper at http://arxiv.org/abs/1110.3113 have
more on this, including some Mathematica programs.
Using them, finding the Wieferich-non-Wilson
primes 2, 3, and 14771 took me only a few minutes. Checking that
there are no others up to 10^7 took Mossinghoff longer.