# Wieferich prime

By Fermat's (Little) Theorem any odd prime*p*divides 2

^{p-1}-1. A prime

*p*is a

**Wieferich prime**if

*p*

^{2}divides 2

^{p-1}-1. In 1909 Wieferich proved that if the first case of Fermat's last theorem is false for the exponent

*p*, then

*p*satisfies this criterion. Since 1093 and 3511 are the only known such primes (and they have been checked to at least 32,000,000,000,000), this is a strong statement!

In 1910 Mirimanoff proved the analogous theorem
for 3 (that the first case of Fermat's last theorem is
false for the exponent *p*, then
*p*^{2} divides 3^{p-1}-1),
but there is little glory in being second. Such
numbers are not called Mirimanoff primes.

Are there infinitely many Wieferich primes? Probably,
but little is known about their distribution. In 1988
J. H. Silverman [Silverman88] proved that the abc-conjecture
implies that for any positive integer *a* > 1, there exists infinitely many primes *p* such that *p*^{2} does not divide *a*^{p-1}-1. But this is a long way from showing there are finitely many Wieferich primes.

**See Also:** WilsonPrime, FermatQuotient, CatalansProblem

**Related pages** (outside of this work)

**References:**

- CDP97
R. Crandall,K. DilcherandC. Pomerance, "A search for Wieferich and Wilson primes,"Math. Comp.,66:217 (1997) 433--449.MR 97c:11004(Abstract available)- Silverman88
J. H. Silverman, "Wieferich's criterion and the abc-conjecture,"J. Number Theory,30:2 (1988) 226--237.MR 89m:11027- Wieferich09
A. Wieferich, "Zum letzten Fermat'schen theorem,"J. Reine Angew. Math.,136(1909) 293--302.

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