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A Wieferich prime search up to $6·7×{10}^{15}$. (English) Zbl 1278.11003
Summary: A Wieferich prime is a prime $p$ such that ${2}^{p-1}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{2}\right)$. 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 $p<6·7×{10}^{15}$. Furthermore, our method allowed for the efficient collection of statistical data on Fermat quotients, leading to a strong empirical confirmation of a conjecture of R. Crandall, K. Dilcher and C. Pomerance [Math. Comput. 66, No. 217, 433–449 (1997; Zbl 0854.11002)]. Our methods proved flexible enough to search for new solutions of ${a}^{p-1}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{2}\right)$ 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 $p<9·7×{10}^{14}$.
##### MSC:
 11-04 Machine computation, programs (number theory) 11A41 Elementary prime number theory 11Y16 Algorithms; complexity (number theory) 11Y11 Primality