We answer the undocumented question of whether there exists a uniform algebraic expression that defines terms in Geometric Progressions (GP) whose ratios are positive integers; if so, whether the said expression defines terms in any GP whose ratios are not integers. As a solution to an easier method of generating large primes, we discover a set whose items are not pseudo-primes to base 2 of Fermat’s primality test. Finally, we discuss the application of the aforementioned method in pseudo-random number generation and in public-key cryptography.

Keywords: Fermat’s primality test, pubic-key cryptography, pseudo-random number generation, PAF algorithm