If P is a prime and 2P+1 is also prime, then P is a Sophie Germain prime. In this article several new Sophie Germain primes are reported, which are the largest known at this time. The search method and the expected search times are discussed.

The numbers 242206083 · 238880 ± 1 are twin primes. The number p = 2375063906985 · 219380 − 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515 · 25056 − 1, the numbers p, p + 2 and 2p + 1 are all primes. In the first days of October, 1995, Harvey Dubner [4] found the largest known twin primes with 5129 decimal digits. (Our earlier twin prime record was 6970538...

a power digraph, denoted by $g(n,k)$, is a directed graph with $z_{n}={0,1,..., n-1}$ as the set of vertices and $l={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. in this paper, the structure of $g(2q+1,k)$, where $q$ is a sophie germain prime is investigated. the primality tests for the integers of the form $n=2q+1$ are established in terms of th...

In the [16] has been proven that the multiplicative group Z/pZ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. M...

In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will be unconditional: in particular, it will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any s...

Since 2005, PrimeGrid has grown from a small project factorising RSA numbers by brute force to one of the largest volunteer computing projects in the world. The project has discovered over 60 new million-digit primes, as well as record sized twin and Sophie Germain primes. This paper will present a history of the project, the algorithms and software used by PrimeGrid and how the BOINC distribut...