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Wieferich primes and Wieferich numbers Siavashi, Sahar

Description

An odd prime $p$ is called a \emph{Wieferich prime} (in base $2$), if $$2^{p-1} \equiv 1 \pmod {p^2}.$$ These primes first were considered by A. Wieferich in $1909$, while he was working on a proof of Fermat's last theorem. This notion can be generalized to any integer base $a>1.$ In this talk, we discuss the work that has been done regarding the size of the set of non-Wieferich primes and show that, under certain conjectures, there are infinitely many non-Wieferich primes in certain arithmetic progressions. Also we consider the congruence $$a^{\varphi(m)} \equiv 1 \pmod{m^2},$$ for an integer $m$ with $(a,m)=1,$ where $\varphi$ is Euler's totient function. The solutions of this congruence lead to Wieferich numbers in base $a$. In this talk we present a way to find the largest known Wieferich number for a given base. In another direction, we explain the extensions of these concepts to other number fields such as quadratic fields of class number one.

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Attribution-NonCommercial-NoDerivatives 4.0 International