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Zbl 1246.11019
Dobeš, Jan; Kureš, Miroslav
Search for Wieferich primes through the use of periodic binary strings. (English)
[J] Serdica J. Comput. 4, No. 3, 293-300 (2010). ISSN 1312-6555

A Wieferich prime is a prime $p$ that satisfies $2^{p-1}\equiv 1 \bmod p^2$. 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 Wieferich\@Home. One of the results of the project is that if $n$ is $B$-periodic of a bit pseudo-length $\leq 3500$ obtained by concatenating a bit string of bit pseudo-length $\leq 24$, and $n+1$ is a Wieferich prime, then $n=p_1-1$ or $n=p_2-1$.
[Thomas Stoll (Vand\oe uvre-lés Nancy)]

MSC 2000:: *11A63 Radix representation
11Y99 Computational number theory

Keywords: Wieferich prime; periodic bit string; distributed computing

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