### Abstract

To generate random numbers (RNs) of long period for large scale simulation studies, the usual multiplicative congruential RN generator can be extended to higher order. A multiplicative congruential RN generator of order two with prime modulus 2^{31} - 1 attains a maximal period of (2^{31} - 1)^{2} - 1 when the two multipliers (a_{1},a_{2}) are chosen properly. By fixing a_{2} at -742938285, a multiplier recommended for first-order generator in a previous study, approximately 1.1 billion choices of a_{1} which are able to produce RNs of maximal period are investigated in this paper. Via the spectral test of dimensions up to six, 14 sets of multipliers (a_{1},a_{2}) exhibit good lattice structure in a global sense with a spectral measure greater than 0.84. Ten of these multipliers also pass a battery of tests for detecting departures from local randomness and homogeneity. Furthermore, the execution time is promising on 32-bit machines. In sum, the second-order generators devised in this paper possess the properties of long period, randomness, homogeneity, repeatability, portability, and efficiency, required for practical use.

Original language | English |
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Pages (from-to) | 113-121 |

Number of pages | 9 |

Journal | Computers and Mathematics with Applications |

Volume | 36 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1998 Aug |

### All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Computers and Mathematics with Applications*,

*36*(3), 113-121. https://doi.org/10.1016/S0898-1221(98)00133-3