  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.
