The Nonzero Imbalance Uniquely Determines Nonequal Prime Pairs

Abstract

We define the imbalance δ(p, q) of primes p and q as δ(p, q) = |p-q| / (p+q) ∈ [0, 1) and prove that, whereas all prime pairs (p, p) have the same imbalance of zero, all pairs of nonequal primes, without loss of generality, (p, q) with p > q, are uniquely determined by their imbalances. Moreover, we find p and q explicitly in terms of those imbalances. Finally, we demonstrate that some rationals in (0, 1) do not constitute valid imbalances for prime pairs and find explicitly what those invalid imbalances are. We connect our results with semiprime factorizations and the Goldbach conjecture.