Problems & Puzzles:
Problems
Problem 59.
WieferichnonWilson primes
Jonathan Sondow sent the following
interesting problems:
Here are some problems on "WieferichnonWilson
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 := ((p1)! + 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^{p1}  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 "WieferichnonWilson 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 WieferichnonWilson prime, i.e., that if p
= 14771, then p divides the "FermatWilson quotient" q_p(w_p)?
(When p = 14771, the number q_p(w_p) has over 800 million
digits.)
Problem
2. Is there a WieferichnonWilson prime greater than
10^7? Are there infinitely many?
Problem
3. Can you prove that infinitely many primes are NOT WieferichnonWilson
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 WieferichnonWilson
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, FermatWilson quotients, and the WieferichnonWilson
primes 2, 3, 14771,

***
On Feb, 8, 2021, Simon Cavegn wrote:
Found no more than the known solutions.
Searched up to 5*10^7.
***
