Random numbers fall mainly in the planes.

The apparently haphazard way in which successive multiplications by a large integer K produce remainders after overflow makes the resulting numbers work surprisingly well for many Monte Carlo problems. Scores of papers have reported favorably on cycle length and statistical properties of such generators. The purpose of this note is to point out that all multiplicative congruential random number generators have a defect-a defect that makes them unsuitable for many Monte Carlo problems and that cannot be removed by adjusting the starting value, multiplier, or modulus. The problem lies in the "crystalline" nature of multiplicative generators-if n-tuples (uu2,,. . . yUn), (u2,u3,... .,un+1))... of uniform variates produced by the generator are viewed as points in the unit cube of n dimensions, then all the points will be found to lie in a relatively small number of parallel hyperplanes. Furthermore, there are many systems of parallel hyperplanes which contain all of the points; the points are about as randomly spaced in the unit n-cube as the atoms in a perfect crystal at absolute zero. One can readily think of Monte Carlo problems where such regularity in "random" points in n-space would be unsatisfactory; more disturbing is the possibility that for the past 20 years such regularity might have produced bad, but unrecognized, results in Monte Carlo studies which have used multiplicative generators. Some Notation.-For any modulus m and multiplier k, let