Pseudo-random number generators based on linear congruence relations using a congruence base of a Mersenne prime (one that is one less than a power of two) and a multiplier obtained from the sum or difference of two powers of two have been proposed. They lead to especially simple and efficient algorithms for bit hacking to generate the sequence.

This paper shows, reasonably straightforwardly, that such choices can lead to sequences of pseudo-random numbers that do not satisfy several desirable properties of random numbers and, hence, counsels against their use in general. The statistical methods used are easy to follow, and the results in specific cases are well illustrated.

This research paper is generally well laid out (if a little cluttered at times), is typographically accurate, and has a short but appropriate set of references. The results extend beyond Mersenne primes to other integers that are a few less than a power of two.

The paper also proposes a solution to the problem it identifies. If multiple generators are used so that the associated recurrence relation is of second or higher order, then the problem disappears. Unfortunately, at this point, the attraction of the simple implementation is weakened. This is a good cautionary tale.