Can a …ctional teenage diary of the mathematician Sophie Germain have dramatic and captivating appeal to audiences ranging from curious teenagers to professional mathematicians? The answer lies in the delicate balance between what we do and don’t know about her real life, along with the extraordinary historical and mathematical circumstances that coalesced with her stranger-than-…ction initiati...

In this paper, we provide a generalization of Proth’s theorem for integers the form Kp^n+1. particular, primality test that requires modular exponentiation (with proper base a) similar to Fermat’s without computation any GCD’s. We also two tests increase chances proving Kp^n+1 primes. As corollaries, three families N whose can be certified only by a^{N-1} \equiv 1 \pmod (Fermat’s test). One the...

A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their product. An examination of the use of Binary Search Trees in implementing the algorithm into a working program is given. In addition an application of the algorith...

There are a number of subsets of primes with a conjectured density of a constant times x/ log x. These include the primes separated by a fixed even integer, Sophie Germain primes and the so-called “thin primes”, i.e. primes of the form 2q − 1 where q is prime [3]. In order to gain some familiarity with sequences of this density we undertook an investigation of the set of primes of prime order, ...

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...

Sophie Germain Counter Mode (SGCM) is an authenticated encryption mode of operation, to be used with 128-bit block ciphers such as AES. SGCM is a variant of the NIST standardized Galois / Counter Mode (GCM) which has been found to be susceptible to weak key / short cycle forgery attacks. The GCM attacks are made possible by its extremely smooth-order multiplicative group which splits into 512 s...

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L / K of squarefree degree n . If E is the normal closure then G = Gal ( ) can be viewed as a permutation group show that has derived length at most 4, but many groups and 2 cannot occur. investigate in detail case where p q ≥ 3 + 1 are both prime. (Thus Sophie Germain prime safeprime). list which aris...

The results of 3 experiments in Costas arrays are presented, for which theoretical explanation is still not available: the number of dots on the main diagonal of exponential Welch arrays, the parity populations of Golomb arrays generated in fields of characteristic 2, and the maximal cross-correlation between pairs of Welch or Golomb arrays generated in fields of size equal to a Sophie Germain ...