The $k$-tuple Prime Difference Champion

TL;DR

This paper investigates the most common prime difference patterns among primes up to a large number, proving their growth and properties unconditionally, and under conjecture, their structure as large square-free numbers containing primorials.

Contribution

It establishes the unbounded growth of $k$-tuple prime difference champions and characterizes their prime factorization properties under a prime tuple conjecture.

Findings

$k$-tuple PDCs tend to infinity.

Unconditionally, $k$-tuple PDCs have asymptotically the same number of prime factors as porimorials.

Under the Hardy-Littlewood conjecture, $k$-tuple PDCs are infinite square-free numbers containing large primorials.

Abstract

Let $D_{k}$ be a set with $k$ distinct elements of integers such that $d_{1}<d_{2}<\cdots<d_{k}$. We say $D_{k}^{*}$ is a $k$-tuple prime difference champion ($k$-tuple PDC) for primes $\le x$ if the set $D_{k}^{*}$ is the most probable differences among $k+1$ primes up to $x$. Unconditionally we prove that the $k$-tuple PDCs go to infinity and further have asymptotically the same number prime factors when weighted by logarithmic derivative as the porimorials. Assuming an appropriate form of the Hardy-Littlewood Prime $k$-tuple Conjecture, we obtain that the $k$-tuple PDCs are infinite square-free numbers containing any large primorial as factor when $x\rightarrow \infty$.