On the Density of Prime Imbalances in the Unit Interval

Abstract

We prove that the set of normalized differences between primes, defined as $S = \{(p-q)/(p+q) : p > q \text{ are primes}\}$, is dense in the open unit interval $(0,1)$. Our proof provides an explicit construction algorithm with quantitative bounds, relying on elementary results from prime number theory including Bertrand's postulate and explicit bounds on prime gaps in long intervals.