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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 $2^{p-1}\equiv 1 \bmod p^2$. 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 $n=p_2-1$.
11A63Radix representation; digital problems
11Y99Computational number theory
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