This paper presents the results of an exhaustive analysis of all of the prime modulus multiplicative congruential random number (RN) generators with moduli smaller than 215. In an amount of around 20 million multipliers which are able to produce a full period of RNs, 239 multipliers have a good lattice structure. Among which 52 multipliers further pass a comprehensive battery of empirical tests. These 52 multipliers thus possess good local and global statistical properties. It is worthwhile to note that some empirically tested multipliers recommended in some previous studies are not on this list. The conclusion is that both theoretical and empirical tests are mandatory to sieve out good multipliers. To generate RNs of very long period, many existing techniques can be applied without further effort.
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