Wieferich primes  find a third Wieferich prime

For elMath.org: Miroslav Kures 
Institute of Mathematics, Brno University of Technology 
Contact: 
Creation date: 18.12.2007 

Find a third Wieferich prime
One of basic and wellknown results of number theory is socalled Little Fermat Theorem: For every prime p and every integer x, which is not a multiple of p, the identity
is satisfied. Let us explain the notation. First, we recall that the prime is a natural number greater than 1 which has no proper divisors; it has only 1 and itself as divisors. The list of primes starts from 2, 3, 5, 7, 11, 13, etc. and it is not difficult to prove that there are infinitely many primes. The notation x^{p−1} ≡ 1 ( mod p) means that if we exponentiate an arbitrary number x to p−1 and if we divide a result by p, then we have obtained 1 as the remainder. We take p=5 and x=3 as an example. Certainly, we evaluate 3^{5−1}=3^{4}=81 and after the division 81/5 we have the remainder 1. Now, we choose x=2. Then Little Fermat Theorem holds for every odd prime, i.e. the equation
is satisfied for infinitely many primes p, in fact for all primes starting from 3. We put the question: are there any primes for which the equation
is satisfied? Once again: after the exponentiation of 2 to (p−1)th power and the subsequent division by the square of p is to be to obtain 1 as the remainder after the division. If it is really so, p is called the Wieferich prime.
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 Last Fermat Theorem says that for n > 2 there do not exist integers a, b, c for which a
^{n}+b
^{n}+c
^{n}=0.) After graduation in the University of Münster, Arthur Wieferich had become the high school teacher and he did not follow a research. He died on the September 15, 1954, in Meppen. His results are deep and valuable. Let us recall the main theorem form the mentioned paper.
Theorem (Wieferich 1909). Let p be an odd prime, and let a,b,c be nonzero integers such that a^{p} + b^{p} + c^{p} = 0. Furthermore, assume that p does not divide the product abc. Then p is a Wieferich prime.
A number of properties of Wieferich primes is proved. Many of them are about relations between Wieferich primes and Mersenne primes. Nevertheless, only two Wieferich primes are known: 1093 (discovered by W. Meissner in 1913) and 3511 (discovered by N. G. W. H. Beeger in 1922). If you want check for the equalities
2^{1092} ≡ 1 ( mod 1194649) and 2^{3510} ≡ 1 ( mod 12327121), 

we call you attention to algorithms for simple exponentiation of "forbidding large" powers.
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 nonWieferich primes. As to literature, there is no third Wieferich prime approximately to 10^{15}.
Author: Miroslav Kures