Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 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.
| Named after | Arthur Wieferich |
|---|---|
| Publication year | 1909 |
| Author of publication | Wieferich, A. |
| No. of known terms | 2 |
| Conjectured no. of terms | Infinite |
| Subsequence of |
|
| First terms | 1093, 3511 |
| Largest known term | 3511 |
| OEIS index | A001220 |
Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.
As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).