# Subjects

## Wieferich primes

Find a third Wieferich prime Published:Miroslav KuresCreation date:2007-12-18Arthur Wieferich was born on 27th April 1884 in MÃ¼nster, Germany. He had published five original papers, four of them (written in 1908 and 1909) turn out to be important for a development in the number theory. In the paper Zum letzten Fermat'schen Theorem, he demonstrates a relation between described primes and the most famous mathematical question (answered by Andrew Wiles, on June 23, 1993), Last Fermat Theorem.

The abc conjecture and non-Wieferich primes Published:Miroslav KuresCreation date:2007-12-20The known Wieferich primes are 1093 and 3511. It is not known, if there exist finitely or infinitely many Wieferich primes; Even it is not known, if there exist finitely or infinitely many non-Wieferich primes. However, we have a result allied to so-called abc-conjecture, which can be formulated for positive integers

a,b,cas follows.

Wieferich primes and Mersenne primes Published:Miroslav KuresCreation date:2007-12-28Let n ∈ N;

Mersenne numbersare defined by M_{n}=2^{n}−1. A binary expression of Mersenne numbers consists of units only. If M_{n}is a prime number, then n is a prime number. (The reverse implication need not hold.) ...

Near Wieferich primes Published:Miroslav KuresCreation date:2008-01-14Let us express numbers (mod

p^{2}) in a formZ+Apwith bothZ,Areduced (modp). Wieferich primes are defined by 2^{p}^{-1}≡ 1 (modp^{2}), i.e. 2^{p}^{-1}=1+0p. Thus, for Wieferich primes...

New progress in searching for Wieferich primes without a positive result. Wieferich@Home carries on the searching process! Published:Miroslav KuresCreation date:2009-02-161093 and 3511 are only two known Wieferich primes up to now. Knauer and Richstein searched for Wieferich primes up to 1.25x10

^{15}with no other Wieferich primes. Their result was published in 2005. Dorais and Klyve searched for Wieferich primes up to 6.7x10^{15}with no new discovery.

Search for Wieferich Primes through the Use of Periodic Binary Strings Published:Project AuthorsCreation date:2010-10-21The result of the distributed computing project Wieferich@Home is presented: the binary periodic numbers of bit pseudo-length j <= 3500 obtained by replication of a bit string of bit pseudo-length k <= 24 and increased by one are Wieferich primes only for the cases of 1092 or 3510.