Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-Non-Wilson Primes 2, 3, 14771

Abstract: The Fermat quotient

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

, for prime

p m i d a

, and the Wilson quotient

w_{p} : = ((p - 1)! + 1) / p

are integers. If

p m i d w_{p},

then

p

is a Wilson prime. For odd

p,

Lerch proved that

(s u m_{a = 1}^{p - 1} q_{p} (a) - w_{p}) / p

is also an integer; we call it the Lerch quotient

e l l_{p} .

If

p m i d e l l_{p}

we say

p

is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to

3 i m e s 1 0^{6}

; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if

p

is a non-Wilson prime, then

q_{p} (w_{p})

is an integer that we call the Fermat-Wilson quotient of

p .

The GCD of all

q_{p} (w_{p})

is shown to be 24. If

p m i d q_{p} (a),

then

p

is a Wieferich prime base

a

; we give a survey of them. Taking

a = w_{p},

if

p m i d q_{p} (w_{p})

we say

p

is a Wieferich-non-Wilson prime. There are three up to

1 0^{7}

, namely, 2, 3, 14771. Several open problems are discussed.