We define a Cayley Table T from the structure (J,+) based on a subset J of Z containing prime numbers and their additive inverses, which we then use as a model of gaps between primes of the form pα − 3. Using definitions of the relationships between primes and their gaps derived from T, we prove the existence of infinitely many pairs of primes, (pn, pn+m), such that (pn+m − pn) = (pα − 3) where n, α ≥ 3 and m ≥ 1 and pn is the nth prime. Finally, we use this result to show the existence of infinitely many pairs of prime numbers with a gap of 2.
MSC Classification: 11N05, 11B05