Wieferich primeBy Fermat's (Little) Theorem any odd prime p divides 2p-1-1. A prime p is a Wieferich prime if p2 divides 2p-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 p2 divides 3p-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 p2 does not divide ap-1-1. But this is a long way from showing there are finitely many Wieferich primes.
Related pages (outside of this work)
- R. Crandall, K. Dilcher and C. Pomerance, "A search for Wieferich and Wilson primes," Math. Comp., 66:217 (1997) 433--449. MR 97c:11004 (Abstract available)
- J. H. Silverman, "Wieferich's criterion and the abc-conjecture," J. Number Theory, 30:2 (1988) 226--237. MR 89m:11027
- A. Wieferich, "Zum letzten Fermat'schen theorem," J. Reine Angew. Math., 136 (1909) 293--302.