On the Density of Prime Imbalances in the Unit Interval

TL;DR

This paper proves that the normalized differences between primes are dense in the interval (0,1), using elementary prime number theory results and providing an explicit construction method.

Contribution

It establishes the density of prime differences in (0,1) and offers an explicit algorithm with quantitative bounds based on elementary prime number results.

Findings

Normalized prime differences are dense in (0,1)

Provides an explicit construction algorithm

Uses elementary prime number bounds

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.