REAMC Report-004

Two Conjectures on Statistical Performance of ACORN Generators: Evidence for Orders 11 - 15

R S Wikramaratna

August 2021


The Additive Congruential Random Number (ACORN) generator represents an approach to generating uniformly distributed pseudo-random numbers which is straightforward to implement for arbitrarily large order and modulus (where the modulus is a sufficiently large power of 2, typically up to 2120); it has been demonstrated in previous papers to give rise to sequences with long period which, for the k-th order ACORN generator with modulus a power of 2, can be proven from theoretical considerations to approximate in a particular defined sense to the desired properties of uniformity in up to k dimensions. REAMC Report-003(2021) recently proposed two conjectures which assert that for order 8 (or larger) and modulus 2120 almost every choice of odd seed together with an arbitrary set of initial values (including the possibility that the initial values are all chosen equal to zero) leads to a different sequence that can reasonably be expected to pass all the tests in the current Version 1.2.3 of the standard empirical test package known as TestU01. Supporting results were included for ACORN generators of modulus 2120 and order 8, 9 and 10. The present report provides further supporting results covering ACORN generators of modulus 2120 and orders 11 to 15 inclusive. The results obtained with increasing order suggest that similar results will continue to be obtained as the order is increased further, with no suggestion as yet of any practical upper limit on the orders for which the conjectures hold.

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