Dependence of two simulated numbers generated by congruences

Overview & Synopsis

When we consider a congruence $ T (x) \equiv ax $ modulo m as a pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that this dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We use this result in order to obtain sequences of random numbers proved IID.