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A Wieferich prime search up to $6.7 \times 10^{15}$. (English) Zbl 1278.11003
Summary: 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 $p < 6.7 \times 10^{15}$. Furthermore, our method allowed for the efficient collection of statistical data on Fermat quotients, leading to a strong empirical confirmation of a conjecture of {\it R. Crandall, K. Dilcher} and {\it C. Pomerance} [Math. Comput. 66, No. 217, 433--449 (1997; Zbl 0854.11002)]. 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 less than $p < 9.7 \times 10^{14}$.

11-04Machine computation, programs (number theory)
11A41Elementary prime number theory
11Y16Algorithms; complexity (number theory)
Full Text: EMIS