Primes of the form 2p+1

Abstract

Arithmetic congruences are derived for the exponents of composite Mersenne numbers. It is known that the 2 p −1 is composite if p is a Sophie Germain prime which is congruent to 3 modulo 4. After verifying the equality of estimates of the density of these primes through refined sieve and proabablistic methods, a set of the arithmetic sequences for the exponents are listed. The coefficients in these sequences generally have a nontrivial common factor, and a shift in the number of doubling cycles for a given number of partitions is found to yield divisible Mersenne numbers. Several sequences with relatively prime coefficients have terms that are congruent to 3 modulo 4. Furthermore, a linear recursion relation for the exponents would have a non-zero density of solutions representing positive integers in the natural numbers by the Skolem-Mahler-Lech theorem, thereby predicting the infinite extent of the prime exponents of composite Mersenne numbers of this kind and the Sophie Germain primes.