What is a Wieferich pair in mathematics?

A Wieferich pair is a pair of prime numbers that satisfy specific modular arithmetic conditions involving each other's squares. Named after the German mathematician Arthur Wieferich, these pairs played a crucial role in Preda Mihăilescu's 2002 proof of Catalan's conjecture, now known as Mihăilescu's theorem. They extend concepts originally associated with individual Wieferich primes, which are connected to number theory and Fermat's theorems.

What we know

• A Wieferich pair consists of a pair of prime numbers p and q satisfying p^q-1 ≡ 1 (mod q^2) and q^p-1 ≡ 1 (mod p^2). [4]
• Wieferich pairs are named after German mathematician Arthur Wieferich, whose earlier work on Fermat's Last Theorem also described single Wieferich primes. [2, 4]
• Wieferich pairs played an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem, formerly known as Catalan's conjecture. [4]

Where to go next

• whoWho was Arthur Wieferich?
• how we knowHow did Mihăilescu prove Catalan's conjecture?
• a case in pointWhat is a Wieferich prime?
• the bigger pictureWhat is Catalan's conjecture?
• compared withWhat are Mersenne and Fermat numbers?
• an unexpected connectionHow do prime pairs connect to Fermat's Last Theorem?Arthur Wieferich originally described Wieferich primes while working on Fermat's Last Theorem, connecting modular properties of primes to centuries-old diophantine problems.

Sources

• [1]Wieferich prime (Wikidata Q966601) · CC0 1.0
• [2]Wieferich prime · CC BY-SA 4.0
• [3]List of prime numbers · CC BY-SA 4.0
• [4]Wieferich pair · CC BY-SA 4.0
• [5]List of unsolved problems in mathematics · CC BY-SA 4.0