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Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. From the definition of the congruence relation on integers , it follows that this property is equivalent to the definition given at the beginning.

The following are two illustrative examples using the primes 11 and Wieferich primes can be defined by other equivalent congruences. The converse is also true: This also implies that Wieferich primes can be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p 2 coincide: He asked the question when this residue vanishes and tried to find expressions for answering this question.

The prime was found to be a Wieferich prime by Waldemar Meissner in and confirmed to be the only such prime below Haentzschel later ordered verification of the correctness of Meissners congruence via only elementary calculations. The prime was first found to be a Wieferich prime by N. Beeger in  and another proof of it being a Wieferich prime was published in by Guy.

It is now known that if any other Wieferich primes exist, they must be greater than 6. In —, a search for Wieferich primes was performed by the distributed computing project Wieferich Home. The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in The above case where p does not divide any of x , y or z is commonly known as the first case of Fermat's last theorem FLTI   and FLTI is said to fail for a prime p , if solutions to the Fermat equation exist for that p , otherwise FLTI holds for p.

Silverman showed in that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by W 2 and W 2 c respectively,  are complementary sets , so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers.

Since Mersenne numbers of prime indices M p and M q are co-prime,. Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free. If q is prime and the Mersenne number M q is not square-free, that is, there exists a prime p for which p 2 divides M q , then p is a Wieferich prime.

Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: Otherwise, this is p times than that. For the primes and , it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because and are neither prime nor powers of 2. The Wieferich Home project searched for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.

Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2 x reduced modulo m. Furthermore, the following was shown: Let p be a Wieferich prime. Furthermore, the following result was obtained: Then q is a Wieferich prime. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes.

The two notions of nearness are related as follows. It's a conjecture that for every natural number a , there are infinitely many Wieferich primes in base a. For more information, see    and. Known solutions for small a are: Here's what you're missing out on! A text message with your code has been sent to: Didn't receive the code?

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