Prime Pair Products Based on Prime Gaps

This paper investigates the arithmetic structure of prime pair products by analyzing the gaps between primes. Focusing on pairs, we explore patterns in the distribution of their products. We establish that prime pairs with gaps a power of 6 yield products in the form 6n+1, while those without power of 6 gaps result in products of the form 6n+5, except where 2 and 3 are found, offering a foundational understanding of their algebraic properties.