will contain infinitely many prime terms — known as Mersenne primes. The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its divisors apart from itself. Euclid pointed out that 2k−1(2k − 1) is perfect whenever 2 − 1 is pr...

Kim et al (Appl Math & Comput 199 (2008) 195) recently presented a test of random number generators based on the gambler’s ruin problem and concluded that several generators, including the widely used Mersenne Twister, have hidden defects. We show here that the test by Kim et al suffers from a subtle, but consequential error: Re-seeding the pseudorandom number generator with a fixed seed for ea...

Abstract: In this paper, we consider norms and spreads of RFMLR circulant matrices involving the Fermat, Mersenne sequences and Gaussian Fibonacci number, respectively. Firstly, we reviewed some properties of the Fermat, Mersenne sequences, Gaussian Fibonacci number and RFMLR circulant matrices. Furthermore, we give lower and upper bounds for the spectral norms and spread of these special matri...

The beauty of these relationships is that, by inspection alone, one may instantly observe whether or not a prime number is = 1, or = 3 (mod 4). These relationships are especially valuable for very large prime numbers-such as the larger Mersenne primes. Thus, it is seen from inspection of the penultimate digits of the Mersenne primes, as given in [1 ] , that all of the given primes are = 3 (mod ...

The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas–Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth’s Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given....