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Search for Wieferich primes through the use of periodic binary strings. (English) Zbl 1246.11019
A Wieferich prime is a prime p that satisfies 2 p-1 1modp 2 . The only known Wieferich primes to date are p 1 =1093 and p 2 =3511. It is not known whether there are finitely or infinitely many Wieferich primes. Wells Johnson (1977) observed that when p 1 -1 and p 2 -1 are represented in base 2, two periodic (B-periodic) words are obtained. The authors present the algorithms and the results of the distributed computing project WieferichHome. One of the results of the project is that if n is B-periodic of a bit pseudo-length 3500 obtained by concatenating a bit string of bit pseudo-length 24, and n+1 is a Wieferich prime, then n=p 1 -1 or n=p 2 -1.
MSC:
11A63Radix representation; digital problems
11Y99Computational number theory