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Fermat Quotient

The Fermat quotient for a number $a$ and a Prime base $p$ is defined as

q_p(a)\equiv {a^{p-1}-1\over p}.
\end{displaymath} (1)

If $p\notdiv ab$, then
$\displaystyle q_p(ab)$ $\textstyle =$ $\displaystyle q_p(a)+q_p(b)$ (2)
$\displaystyle q_p(p\pm 1)$ $\textstyle =$ $\displaystyle \mp 1$ (3)
$\displaystyle q_p(2)$ $\textstyle =$ $\displaystyle {1\over p} \left({1-{1\over 2}+{1\over 3}-{1\over 4}+\ldots-{1\over p-1}}\right),$ (4)

all (mod $p$). The quantity $q_p(2)=(2^{p-1}-1)/p$ is known to be Square for only two Primes: the so-called Wieferich Primes 1093 and 3511 (Lehmer 1981, Crandall 1986).

See also Wieferich Prime


Crandall, R. Projects in Scientific Computation. New York: Springer-Verlag, 1986.

Lehmer, D. H. ``On Fermat's Quotient, Base Two.'' Math. Comput. 36, 289-290, 1981.

© 1996-9 Eric W. Weisstein