Title

A Wieferich Prime Search up to 6.7 × 10^15

Authors

François G. Dorais, The University Of Michigan
Dominic Klyve, Central Washington UniversityFollow

Document Type

Article

Department or Administrative Unit

Mathematics

Publication Date

10-6-2011

Abstract

A Wieferich prime is a prime p such that 2^p-1 ≡ 1 (mod p²). 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 × 10¹⁵. 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 ≡ 1 (mod p²) 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 × 10¹⁴.

Comments

This article was originally published in Journal of Integer Sequences. The full-text article from the publisher can be found here.

Due to copyright restrictions, this article is not available for free download from ScholarWorks @ CWU.

Recommended Citation

Dorais, F., & Klyve, D. (2011). A Wieferich Prime Search up to 6.7 × 10¹⁵. Journal of Integer Sequences, 14.

Journal

Journal of Integer Sequences

Copyright

© The Author(s) 2011