R S Wikramaratna
May 2023
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) and Report - 004(2021) 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 in Report - 003 (for order 8, 9 and 10) and in Report - 004 (for orders 11 to 15 inclusive).
The present report provides further supporting results covering ACORN generators of modulus 2120 and selected orders between 16 and 101. The cases reported here use the same set of 1000 seed values as in the two earlier reports; there is no reason to suspect that the statistical behaviour with other orders in the same range will be any different to the cases reported here. The results obtained with increasing order suggest that similar results would 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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