The $k$-tuple Prime Difference Champion

Abstract

Let $D_k$ be a set with $k$ distinct elements of integers such that $d_1<d_2<\cdots <d_k$. We say $D_k^{\ast }$ is a $k$-tuple prime difference champion ($k$-tuple PDC) for primes $\le x$ if the set $D_k^{\ast }$ 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 $x\to \infty$.

Some small revise