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Searching for Large-Order Multiple Recursive Generators

Pseudo-random numbers (PRNs) are the basis for almost any statistical simulation and this depends largely on the quality of the pseudo-random number generator (PRNG) used. In this study, we used some results from number theory to propose an efficient method to accelerate the computer search of super-order maximum-period multiple recursive generators (MRGs). We conduct efficient computer searches and identify many efficient and portable MRGs of super-orders, 40751, 50551, and 50873; which respectively have equi-distribution property up to 40751, 50551, and 50873 dimensions, and period lengths of approximately 10380278.1, 10471730.6, and 10474729.3. Using the generalized Mersenne prime algorithm, we extend some known results of some efficient, portable and maximum-period MRGs. In particular, the DX/DL/DS/DT large-order generators are extended to super-order generators. An extensive empirical evaluation shows that these generators behave well when tested with the stringent Small Crush and Crush batteries of the TestU01 package.