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This monograph proves that any finite random number sequence is represented by the multiplicative congruential (MC) way. It also shows that an MC random number generator (d, z) formed by the modulus d and the multiplier z should be selected by new regular simplex criteria to give random numbers an excellent disguise of independence. The new criteria prove further that excellent subgenerators (d1,z1) and (d2,z2) with coprime odd submoduli d1 and d2 form an excellent combined generator (d = d1d2,z) with high probability by Sunzi’s theorem of the 5th-6th centuries (China), contrasting the fact that such combinations could never be found with MC subgenerators selected in the 20th-century criteria. Further, a combined MC generator (d = d1d2,z) of new criteria readily realizes periods of 252 or larger, requiring only fast double-precision arithmetic by powerful Sunzi’s theorem. We also obtain MC random numbers distributed on spatial lattices, say two-dimensional 4000 by 4000 lattices which may be tori, with little pair correlations of random numbers across the nearest neighbors. Thus, we evade the problems raised by Ferrenberg, Landau, and Wong.
Random numbers on computers seem to live in a world of integers; Professor Knuth thus begins his monograph The Art of Computer Programming with such descriptions. And this is, in fact, an accurate embodiment of the prophet of Professor Ito. This monograph will convince you how such apparent inconveniences of the restriction to integers can show their own exquisite structures of conveniences, with the magical power of Sunzi’s theorem.
We restrict ourselves to problems of random numbers on computers. We are happy to see many simplifications. Numbers on computers are essentially integers in various sense. We thus need only integer sequences placed on discrete time points; the setting gives random numbers as the outcome of a huge dice thrown in computers at discrete times. Yet, we have still to discuss that the dice is fair and the throwing is not deceitful. We present here what we have found within this restricted circumstance. You will be surprised to see that Numbers, which existed from the beginning of this universe, seem to have prepared neat answers to the present computer problems