Many important well-known constants such as π and ζ(3) can be approximated by a truncated hypergeometric series. A modular algorithm based on rational number reconstruction was previously proposed to reduce space complexity of the well-known binary splitting algorithm [1]. In this paper, we examine some variations of this algorithm using Mersenne number moduli and Montgomery multiplication. Imp...

Given g ∈ N, what is the number of numerical semigroups S = 〈a, b〉 in N of genus |N \ S| = g? After settling the case g = 2 for all k, we show that attempting to extend the result to g = p for all odd primes p is linked, quite surprisingly, to the factorization of Fermat and Mersenne numbers.

In this part, which constitutes a pendent to the part dedicated Mersenne numbers, I continue discuss fantastic contributions towards solution o classical problems of number theory achieved over last decades with use computers. Specifically, address primality testing, factorisations and search prime divisors numbers certain special form, primarily Fermat their friends relations, such as generali...

The Lucas-Lehmer (LL) test is the most efficient known for deterministically testing the primality of Mersenne numbers, i.e., the integers Ml = 2 l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range...

Molecular visualisation algorithms present simplified views of molecules, which allow chemists to gain insight into their structure and function. Many such algorithms are currently in use, including the ribbon diagrams by Richardson [12], which were developed specifically for viewing proteins. None are customised to the visualisation of carbohydrates – in particular, the rings of atoms formed w...

Reconfigurable computing offers an attractive solution to accelerating infrared scene simulations. In infrared scene simulations, the modeling of a number of atmospheric and optical phenomena like scintillation, refraction, blurring due to lens optics and photon noise may be implemented in parallel. All of these require simultaneous and continual generation of random numbers. Furthermore, rando...

The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...