R S Wikramaratna
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.
Extensive empirical testing using standard test software has demonstrated the excellent statistical performance of the ACORN generators with appropriately chosen order and modulus, over a very wide range of initialisations. In this paper we propose two conjectures concerning appropriate initialisations for some specific choices of order and modulus. The conjectures are based on the results of extensive testing that has been reported either in Part 1 of this report or in earlier publications. New results are presented here to support the assertions made in the conjectures - 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.
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The author believes that comparable results and conclusions can be expected for the ACORN generators with any other similar empirical tests that may exist today or may be developed in the future for uniformly distributed pseudo-random numbers (although if the tests are made more demanding then it is possible that the minimum values of the order and modulus may need to be increased in order to fully satisfy the new tests).