Wieferich primes and Mersenne primes

1 Introduction
Let n ∈ N; Mersenne numbers are defined by M_{n}=2^{n}−1. A binary expression of Mersenne numbers consists of units only.
Proposition 1 If M_{n} is a prime number, then n is a prime number. (The reverse implication need not hold.)
Let n=kl be a composite number. Then 2^{n}−1=2^{kl}−1=(2^{l}−1)(2^{l(k−1)}+2^{l(k−2)}+...+2^{l}+1), i.e. 2^{n}−1 is a composite number, too. On the other hand, 2^{11}−1=2047=23·89.
Thus, we monitor especially Mersenne numbers M_{p}, where p is a prime number. If such M_{p} is also a prime, we call it the Mersenne prime. It is not known, how many Mersenne primes exist not even if it is a finite number. Up to now, 44 Mersenne primes are known, the latest one is M_{32582657}, it was discovered in 2006 and it represents a greatest known prime at all.
2 Perfect numbers
A number n ∈ N is called a perfect number if it is equal to the sum of all its divisors, excluding n itself. (For example, 6 and 28 are perfect numbers.)
Proposition 2 n is an even perfect number if and only if it has form n=2^{p−1}(2^{p}−1) and 2^{p}−1=M_{p} is a (Mersenne) prime. (It is not known whether or not there exists an odd perfect number.)3 Are Mersenne numbers square free?
We call a Mersenne number M_{n} divisible by square, if there exists a prime q such that q^{2} divides M_{p}. If there is no such a prime, we call M_{n} square free. If n is a composite number, then M_{n} can be divisible by square, e.g. for n=6 we have M_{6}=63=3^{2}·7. It is remarkable that M_{1092} is divisible by 1093^{2} and M_{3510} is divisible by 3511^{2}, see [1]. (1093 and 3511 are Wieferich primes satisfying 2^{q−1} ≡ 1 ( mod q^{2}).)
If p is a prime, then the question if M_{p} is square free is open. However, the following result is known.
Proposition 3 Let p,q be primes. If q^{2} divides M_{p}, then q is a Wieferich prime.
Evidently, p and q must be odd. Let q^{2} divides M_{p}, i.e.


It follows from this equation, that p must be the order of 2 (in F_{q}), because p is a prime. From the Little Fermat Theorem follows that

and it means that p divides q−1. However, q−1 is even and that is why q−1=2kp for some k ∈ N. Hence

after the exponentiation to the 2kth power

it means q is a Wieferich prime. Nevertheless, it was proved that squares of Wieferich primes 1093 and 3511 never divide M_{p}. Therefore aspirants for it are only squares of possible new Wieferich primes.
4 Wieferich primes and ECC
The interesting application in elliptic curve cryptography is indicated by the following theorem from [2].
Proposition 4 For a Mersenne prime q=M_{p}=2^{p}−1, binomials x^{p}+2^{s} and x^{p}−2^{s}, where s ≠ 0 ( mod p), is irreducible in F_{q}[x] if p is not a Wieferich prime.
References
 [1]
 Guy, R. K., The primes 1093 and 3511, The Mathematics Student 35 (1967), 204206
 [2]
 Baktir, S., Sunar, B., Frequency domain finite field arithmetic for elliptic curve cryptography, preprint
Author: Miroslav Kures