Problems & Puzzles: Problems

Problem 59.  Wieferich-non-Wilson primes

Jonathan Sondow sent the following interesting problems:

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 and my paper at 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.

On December 12, 2014, Jonathan Sondow sent the following related reference to this problem:

Jonathan Sondow,
Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771,


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