The Fermat quotient for a number and a prime base is defined as
(1)
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If , then
(2)
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(3)
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(mod ), where the modulus is taken as a fractional congruence.
The special case is given by
(4)
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(5)
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(6)
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(7)
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(8)
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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).