High-performance streams of (pseudo) random numbers are crucial for the efficient implementation for countless stochastic algorithms, most importantly, Monte Carlo simulations and molecular dynamics simulations with stochastic thermostats. A number of implementations of random number generators has been discussed for GPU platforms before and some generators are even included in the CUDA support...

Abstract The polar code is a unique coding approach that can achieve Shannon's capacity in modern communication systems' discrete memory-less channels with superior reliability, but it not secure enough under attacks for such systems. This study aims to offer comprehensive secured scheme uses combination of and the Mersenne-Twister pseudo-random number generator (MT-PRNG) super encoding. pre-sh...

Let l ≥ 3 be a prime, and let p = 2 − 1 be the corresponding Mersenne number. The Lucas-Lehmer test for the primality of p goes as follows. Define the sequence of integers xk by the recursion x0 = 4, xk = x 2 k−1 − 2. Then p is a prime if and only if each xk is relatively prime to p, for 0 ≤ k ≤ l − 3, and gcd(xl−2, p) > 1. We show, in the first section, that this test is based on the successiv...

Prime numbers have been a source of fascination for mathematicians since antiquity. The proof that there are infinitely many prime numbers is attributed to Euclid (fourth century B.C.). The basic method of determining all primes less than a given number N is the sieve of Eratosthenes (third century B.C.). Diophantus (third (?) century A.D.) was occupied with finding rational number solutions to...

We prove a linear recursion for the generalized Catalan numbers Ca(n) := 1 (a−1)n+1 ( an n ) when a ≥ 2. As a consequence, we show p |Cp(n) if and only if n 6= p−1 p−1 for all integers k ≥ 0. This is a generalization of the well-known result that the usual Catalan number C2(n) is odd if and only if n is a Mersenne number 2 k − 1. Using certain beautiful results of Kummer and Legendre, we give a...

Concepts and implementation of the Cuba library for multidimensional numerical integration are elucidated. 1. Overview Cuba [1] is a library for multidimensional numerical integration. It features four integration algorithms with interfaces for Fortran, C/C++, and Mathematica. All four can integrate vector integrands and their invocation is quite similar to make it easy to switch routines for c...