# François G. Dorais

### A Wieferich prime search up to $6.7\times10^{15}$

##### By F. G. Dorais and D. Klyve
###### Journal of Integer Sequences 14 (2011), no. 9, art. 11.9.2, 14 p.

A Wieferich prime is a prime $p$ such that $2^{p-1} \equiv 1 \pmod{p^2}$. Despite several intensive searches, only two Wieferich primes are known: $p = 1093$ and $p = 3511$. This paper describes a new search algorithm for Wieferich primes using double-precision Montgomery arithmetic and a memoryless sieve, which runs significantly faster than previously published algorithms, allowing us to report that there are no other Wieferich primes $% $. Furthermore, our method allowed for the efficent collection of statistical data on Fermat quotients, leading to a strong empirical confirmation of a conjecture of Crandall, Dilcher, and Pomerance. Our methods proved flexible enough to search for new solutions of $a^{p-1} \equiv 1 \pmod{p^2}$ for other small values of $a$, and to extend the search for Fibonacci-Wieferich primes. We conclude, among other things, that there are no Fibonacci-Wieferich primes $% $.