The class of generalized feedback shift register (GFSR) random number generators was a promising method for random number generation in the 1980's, but was abandoned because of some flaws such as poor performance on certain tests for randomness. The poor performance may be due to the choice of primitive polynomials used in the generators, rather than inherent flaws in the method. The original GFSR generators were all based on primitive trinomials. This project examines several alternative choices of primitive polynomials with more than one "interior" term to address this problem and hopefully provide access to good random number generators.