The Fermat quotient for a number and a prime base
is defined as
(1)
|
If ,
then
(2)
| |||
(3)
|
(mod ),
where the modulus is taken as a fractional congruence.
The special case
is given by
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
|
all again (mod )
where the modulus is taken as a fractional congruence,
is the digamma
function, and the last two equations hold for odd primes
only.
is an integer for
a prime, with the values for
, 3, 5, ... being 1, 3, 2, 5, 3, 13, 3, 17, 1, 6, ....
The quantity
is known to be congruent to zero (mod
) for only two primes: the
so-called Wieferich primes 1093 and 3511 (Lehmer
1981, Crandall 1986).