Coding the Lehmer pseudo-random number generator

An algorithm and coding technique is presented for quick evaluation of the Lehmer pseudo-random number generator modulo 2 ** 31 - 1, a prime Mersenne number which produces 2 ** 31 - 2 numbers, on a p -bit (greater than 31) computer. The computation method is extendible to limited problems in modular arithmetic. Prime factorization for 2 ** 61 - 2 and a primitive root for 2 ** 61 - 1, the next largest prime Mersenne number, are given for possible construction of a pseudo-random number generator of increased cycle length.