Prime Difference Champions

TL;DR

This paper investigates the most frequent prime differences up to a large number, showing under certain conjectures they align with primorials, and provides unconditional results about their growth and prime factorization properties.

Contribution

It proves that prime difference champions tend to be primorials under the Hardy-Littlewood conjecture and establishes unconditional growth and factorization properties.

Findings

PDCs align with primorials assuming Hardy-Littlewood conjecture.

Unconditionally, PDCs tend to infinity.

PDCs have asymptotically the same number of prime factors as primorials.

Abstract

A Prime Difference Champion (PDC) for primes up to $x$ is defined to be any element of the set of one or more differences that occur most frequently among all positive differences between primes $\le x$. Assuming an appropriate form of the Hardy-Littlewood Prime Pair Conjecture we can prove that for sufficiently large $x$ the PDCs run through the primorials. Numerical results also provide evidence for this conjecture as well as other interesting phenomena associated with prime differences. Unconditionally we prove that the PDCs go to infinity and further have asymptotically the same number of prime factors when counted logarithmically as the primorials.