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Zbl pre05977305
Dorais, François G.; Klyve, Dominic
A Wieferich prime search up to $6.7 \times 10^{15}$. (English)
[J] J. Integer Seq. 14, No. 9, Article 11.9.2, 14 p., electronic only (2011). ISSN 1530-7638/e

Summary: A Wieferich prime is a prime $p$ such that $2^{p-1} \equiv 1 (\bmod \, p^{2})$. 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 $a^{p-1} \equiv 1 (\bmod \, p^{2})$ 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 $p < 9.7 \times 10^{14}$.

MSC 2000:: *11A41 Elementary prime number theory
11Y16 Algorithms
11Y11 Primality

Keywords: Wieferich prime; Fibonacci-Wieferich prime; Wall-Sun-Sun prime; wheel sieve; magic sieve; Montgomery arithmetic

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