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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 1(modp 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<6·7×10 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 1(modp 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<9·7×10 14 .
MSC:
11-04Machine computation, programs (number theory)
11A41Elementary prime number theory
11Y16Algorithms; complexity (number theory)
11Y11Primality