The k-tuple prime difference champion

Abstract

Let Dk be a set with k distinct elements of integers such that d1<d2<⋯<dk. We say that the set Dk⁎ is a k-tuple prime difference champion (PDC) for primes ≤x if Dk⁎ 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 of prime factors when weighted by logarithmic derivative as the primorials. 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→∞.