Fermat quotient (another Prime Pages' Glossary entries) Glossary: Prime Pages: Top 5000: GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

By Fermat's Little Theorem, the quotient (ap-1-1)/p must be an integer. This integer (here denoted qp(a)) is the Fermat quotient of p (with base a). Below are just a few of the nice properties of these numbers.

• qp(ab) = qp(a) + qp(b) (mod p)
• qp(p-1) = 1 (mod p)
• qp(p+1) = -1 (mod p)
• -2 qp(2) = 1 + 1/2 + 1/3 + 1/4 + ... + 1/((p-1)/2) (mod p)
(Eisenstein proved all of these in 1850.)

Finally, note that qp(a)=0 (mod p) is the same as requiring

ap-1 = 1 (mod p2).
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).
Solutions of ap-1 = 1 (mod p2) for odd prime bases a
aValues of p
311, 1006003
520771, 40487, 53471161, 1645333507, 6692367337, 188748146801
75, 491531
11 71
13 863, 1747591
17 3, 46021, 48947
19 3, 7, 13, 43, 137, 63061489

Related pages (outside of this work)

References:

Granville87
A. Granville, "Diophantine equations with varying exponents," Ph.D. thesis, Queen's University in Kingston, (1987)
Keller98
W. Keller, "Prime solutions p of ap-1≡ (mod p2) for prime bases a," Abstracts Amer. Math. Soc., 19 (1998) 394.
Ribenboim83
P. Ribenboim, "1093," Math. Intelligencer, 5:2 (1983) 28--34.  MR 85e:11001 [(Lists many nice properties of these numbers)]
Ribenboim95 (pp. 335-6, 345-9)
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]