In thispaper we discuss properties of generalized Mersenne numbers of the form M(a,p) =ap-1a-1,and proposes a algorithm for searching primes of the form M(a,p).

In this paper we combine the generalized Mersenne prime and the CM method to construct Montgomery-form elliptic curves, besides it sufficiently use pre-computation, compared with the former method, the new method not only improves the speed of selecting curves, but also improves the efficiency of the public key cryptosystems.

Lucas and Lehmer gave a classical primality test for Mersenne numbers and Benedict presented in a recent paper("An elliptic curve test for Mersenne primes",Journal of Number Theory,2005,110(1),pp.

Let P is an odd prime, in this paper we prove that the maximal square free number Q(2p-1) of Mersenne numbers 2p-1 satisfies:Q(2p-1)≥min (2p-1, (πP/logp)2

Nearly all Mersenne numbers and Fermat numbers are Primes

Let p be a prime Then the positive integer M p=2 p-1 is called a Mersenne number In this paper we discuss the lower bounds for the square free part, the greatest prime factor and the number of distinct prime factors of M p .

This paper extends the derivatives of binary sequences of reference [1] in two different ways and defines two different generalized derivatives of binary sequences.

In this paper, we gave the generalized derivative definition of mapping at infinitely space and took the derivative intead of the Frechet derivative of smooth mapping.

Then the periodicity of the generalized derivatives of periodic binary sequences is studied and some properties of the generalized derivatives are provided.