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<caldwell@utm.edu> By Fermat's Little Theorem, the quotient (a^{p1}1)/p must be an integer. This integer (here denoted q_{p}(a)) is the Fermat quotient of p (with base a). Below are just a few of the nice properties of these numbers.
Finally, note that q_{p}(a)=0 (mod p) is the same as requiring a^{p1} = 1 (mod p^{2}).The case a=2 is the Wieferich primes. Below we list several examples of solutions to this congruence from Wilfrid Keller and Jörg Richstein's web page (also linked below). Before Wiles proved Fermat's Last Theorem in 1995, this congruence provided the most powerful tool for proving the first case. Wieferich proved in 1909 that if FLT holds for p, then it must satisfy this congruence with a=2. In 1910 Mirimanoff extended this to the case a=3. As time went on, this was extended up through a=89 [Granville87] (this is enough to show that the first case of FLT is false for all exponents n less than 23,270,000,000,000,000,000).
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Chris K. Caldwell © 19992014 (all rights reserved)
