A step towards proving de Polignac's Conjecture

Abstract: Consider the set of all natural numbers that are co-prime to primes less than or equal to a given prime. Then given a consecutive pair of numbers in that set with an arbitrary even gap, we prove there exists an unbounded number of actual prime pairs with that same gap. This conditional proof of de Polignac's conjecture constitutes a proof for a range of known gaps, but the full conjecture requires additional proof that such number pairs exist for all even gaps.