Pseudo-random numbers - A new proposal for the choice of multiplicators

Pseudo-random numbers are usually generated by multiplicative methods. For binary computers the sequences yi+1e{cyrillic}a yi (mod 2k) are common and the derived numbers xi=yi/2k are taken as samples from the uniform distribution in (0, 1). In this paper 4 a ≈2k ξ is proposed as a guide line for the choice of the multiplicator a where ξ is the golden section number {Mathematical expression}. Such values of the factor a have the property that an approximate knowledge of yi will not yield information about the successor yi+1. Bounds for the autocorrelations of the entire sequences are derived. These are of the same order of magnitude as Greenberger's bounds in the case {Mathematical expression}. However, the precise evaluation of the serial correlations for k≤100 indicates that the factors 4 a ≈2k ξ are superior. One million numbers of a special sequence were tested statistically. The included ALGOL and FORTRAN subroutines will enable programmers to make practical use of this paper.