Random Number Generators on Computers

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. 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. Cover Half Title Title Page Copyright Page Table of Contents Preface Chapter 1: Basic Concepts and Tools 1.1: Random Numbers on Computers as a Sample Process 1.2: Integer Arithmetic with a Modulus d 1.3: Arithmetic Structures of T-Periodic Integer Sequences Chapter 2: Group Structures 2.1: Reduced Residue Class Groups 2.2: Orders, Subgroups and the Theorem of Lagrange 2.3: Cyclic Subgroups and Cyclic Groups Chapter 3: Designs of MC Generators 3.1: Periods of MC Generators with Prime Moduluses 3.2: Sophie Germain Primes and Naoya Nakazawa Primes 3.3: Sweep over Relevant Multipliers under SG and NN Primes 3.4: Composite Moduluses and Sunzi’s Theorem Chapter 4: Lattice Structures 4.1: Lattices Accompanied by MC Random Numbers 4.2: Basis Vectors of a Lattice 4.3: Dual Lattices for Spectral Tests 4.4: Spectral Tests of MC (d, z) Lattices Chapter 5: Regular Simplexes and Regular Lattices 5.1: Construction of Regular Simplexes and Regular Lattices 5.2: Actual Lattices Associated with MC Generators 5.3: Regular Simplex Criterions 5.4: Spectral Tests on Regular Simplex Criterions 5.5: Edge Tests on Regular Simplex Criterions of MC Generators Chapter 6: Extended Second Degree Tests 6.1: Second Degree Tests Revisited 6.2: Extended (Generalized) Second Degree Spectral Tests Chapter 7: Three Excellent MC Generators 7.1: The Present Best MC Generator #001 7.2: Prime-Primitive Root Generator #M001 7.3: A Large-Scale Excellent MC Generator #003 Chapter 8: Distribution of MC Random Numbers on Spatial Lattices 8.1: Distribution of MC Random Numbers on Spatial Lattices 8.2: Distribution of Random Numbers on Lattices as a Variable Base Representation of Integers 8.3: Random Root Function on Lattices 8.4: Non-Periodic Tuning of Random Vector Function 8.5: Relevant MC Generators in Non-Periodic Tuning for Random Initial Value Problems 8.6: Procedures of Non-Periodic Tuning 8.7: Periodic Tuning for Random Initial Value Problems 8.8: Resumé for Ferrenberg-Landau-Wong Tests 8.9: Disclosure: Some Periodically Tuned Data 8.9.1: The MC Generator #003 on a Small Spatial Torus 8.9.2: Generator #001 Requires Different Toruses 8.9.3: Spatial Lattice Close to a Square with #001 Generator Chapter 9: Random Number Fields on Time-Space Lattices Closing Hirakata Ransu Factory (HRF) Index