Wieferich primes

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A Wieferich prime is a prime number

p

such that

p 2

divides

2 p  − 1  −  1

, therefore connecting these primes with Fermat’s little theorem, which states that every odd prime

p

divides

2 p  − 1  −  1

. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat’s last theorem, at which time both of Fermat’s theorems were already well known to mathematicians.

Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511 (see A001220). The next candidate is beyond 1.45 × 10 17 (see A001220).

Sequences

A001220 Wieferich primes: primes

p

with the property that

p 2

divides

2 p  − 1  −  1

.

{1093, 3511, ?}