Geodesic
A Wieferich prime search up to \(6.7 \times 10^{15}\)
Journal of integer sequences, Tome 14 (2011) no. 9
A Wieferich prime is a prime $p$ such that $2^{p-1} \equiv 1 (mod p^{2})$. 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 $p 6.7 \times 10^{15}$. 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 $a^{p-1} \equiv 1 (mod p^{2})$ 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 less than $p 9.7 \times 10^{14}$.
Classification : 11A41, 11Y16, 11Y11
Keywords: wieferich prime, Fibonacci-wieferich prime, wall-Sun-Sun prime, wheel sieve, magic sieve, Montgomery arithmetic 
@article{JIS_2011__14_9_a0,
     author = {Dorais,  Fran\c{c}ois G. and Klyve,  Dominic},
     title = {A {Wieferich} prime search up to \(6.7 \times 10^{15}\)},
     journal = {Journal of integer sequences},
     year = {2011},
     volume = {14},
     number = {9},
     zbl = {1278.11003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_9_a0/}
}
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Dorais,  François G.; Klyve,  Dominic. A Wieferich prime search up to \(6.7 \times 10^{15}\). Journal of integer sequences, Tome 14 (2011) no. 9. http://geodesic.mathdoc.fr/item/JIS_2011__14_9_a0/