One method for generating random numbers (RNs) of long period recommended by many scholars is the multiple recursive generator (MRG), in that the current RN is essentially a linear combination of the k preceding ones. In this paper, the upper bounds for a figure of merit adopted in the spectral test are derived for the k th order MRG with p ≤ k terms. As p gets smaller, the bounds become smaller as well. The simplest form of the k th order MRG with two terms frequently discussed in literature is found to have the worst bound.