Subjects

Wieferich primes

Find a third Wieferich prime
Published: Miroslav Kures
Creation date: 2007-12-18

Arthur Wieferich was born on 27th April 1884 in Münster, 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 Kures
Creation date: 2007-12-20

The 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,c as follows.
Wieferich primes and Mersenne primes
Published: Miroslav Kures
Creation date: 2007-12-28

Let n ∈ N; Mersenne numbers are defined by Mn=2n−1. A binary expression of Mersenne numbers consists of units only. If Mn is a prime number, then n is a prime number. (The reverse implication need not hold.) ...
Near Wieferich primes
Published: Miroslav Kures
Creation date: 2008-01-14

Let us express numbers (mod p2) in a form Z+Ap with both Z, A reduced (mod p). Wieferich primes are defined by 2p-1 ≡ 1 (mod p2), i.e. 2p-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 Kures
Creation date: 2009-02-16

1093 and 3511 are only two known Wieferich primes up to now. Knauer and Richstein searched for Wieferich primes up to 1.25x1015 with no other Wieferich primes. Their result was published in 2005. Dorais and Klyve searched for Wieferich primes up to 6.7x1015 with no new discovery.
Search for Wieferich Primes through the Use of Periodic Binary Strings
Published: Project Authors
Creation date: 2010-10-21

The 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.