The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2r − 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give ...

Basic uniform pseudo-random number generators are implemented on ATI Graphics Processing Units (GPU). The performance results of the realized generators (multiplicative linear congruential (GGL), XOR-shift (XOR128), RANECU, RANMAR, RANLUX and Mersenne Twister (MT19937)) on CPU and GPU are discussed. The obtained speed-up factor is hundreds of times in comparison with CPU. RANLUX generator is fo...

Multiplicative congruential random number generators of the form sn = a*Sn_i mod m using the Mersenne prime modulus 2-1 are examined. Results show that they can provide sufficiently long pseudo-random sequences that can be implemented efficiently using 64 bit accumulators without the need of a costly division operation. INTRODUCTION Random number generators are widely used in computer simulatio...

These notes correspond to a mini-course taught by the author during the program “PIMS-SFU undergraduate summer school on multiple zeta values: combinatorics, number theory and quantum field theory”. Please send any comments or corrections to the author at [email protected]. 1. Primes, Mahler Measure, and Lehmer’s question We start our study by discussing prime numbers. First consider the ...