What is a Wieferich prime?

A Wieferich prime is a rare type of prime number that satisfies a specific condition related to Fermat's little theorem, where its square divides two to the power of the prime minus one, minus one. First described by Arthur Wieferich in 1909, only two such primes are known to exist. These special primes have since been linked to various mathematical topics, including Fermat's Last Theorem and the abc conjecture.

What we know

• A Wieferich prime is a prime number p such that p squared divides two to the power of p minus one, minus one, connecting these primes with Fermat's little theorem. [1]
• Arthur Wieferich first described Wieferich primes in 1909 in works pertaining to Fermat's Last Theorem. [1]
• As of 2026, the only known Wieferich primes are 1093 and 3511. [1]
• By Euclid's theorem, there is an infinite number of prime numbers from which subsets like Wieferich primes may be studied. [1, 2]

Where to go next

• whoWho was Arthur Wieferich?
• how it workedHow does Fermat's little theorem work?
• a case in pointWhat is a Wieferich pair?
• compared withWhat is a Wall-Sun-Sun prime?
• what followedHow do these primes connect to the abc conjecture?
• an unexpected connectionHow do primes connect to Mihăilescu's theorem?Wieferich-related pairs play an important role in the proof of Mihăilescu's theorem, formerly known as Catalan's conjecture.

Sources

• [1]Wieferich prime · CC BY-SA 4.0
• [2]List of prime numbers · CC BY-SA 4.0
• [3]Wall–Sun–Sun prime · CC BY-SA 4.0
• [4]Wieferich pair · CC BY-SA 4.0

Your path

1. What was the Ostend Company?
2. /What is a Wieferich prime?