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# The Structure of Linear Congruential Sequences

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## Abstract

This chapter discusses the structure of linear congruential sequences and their suitability as a source of random integers in a computer. Every congruential sequence is made up of a block of t < m residues, the effective period of the sequence, followed by translates of that block. A formula is given for the period, effective period, and translating constant for every sequence. Points in n-space, produced by a congruential generator, fall on a lattice with unit-cell volume m n-1. A congruential random number generator uses a linear transformation on the ring of reduced residues of some modulus to produce a sequence of integers. These integers are converted to fractions of the modulus and serve as independent uniform random variables in Monte Carlo calculations. Given a sequence of integers produced by a random number generator, points are formed in n-space, whose coordinates are successive n-tuples produced by the generator. The lattice spanned by that set of points is called the n-lattice of the generator.