A Power Law Governing Prime Gaps

Abstract

A prime gap is the difference between two successive prime numbers. Prime gaps are casually thought to occur randomly. However, the “k-tuple conjecture” suggests that prime gaps are non-random by estimating how often pairs, triples and larger groupings of primes will appear. The k-tuple conjecture is yet to be proven, but a very recent work presents a result that contributes to a confirmation of thek-tuple conjecture by finding unexpected biases in the distribution of consecutive primes. Here, we present another contribution to confirmation of the k-tuple conjecture based on statistical physics. The pattern we find comes in the form of a power law in the distribution of prime gaps. We find that prime gaps are proportional to the inverse of the chance of a number to be prime.