The K-Tuple Prime Difference Champion

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 for primes≤x if Dk*is the most probable differences among k＋1 primes up to real number x.Suppose d*is the greatest common divisor of k-tuple prime difference champion.Unconditionally we prove that d*goes to infinity and further has asymptotically the same number of prime factors when weighted by logarithmic derivative as the primorials.Assuming an appropriate form of Hardy-Littlewood Prime k-Tuple Conjecture,we obtain that the k-tuple prime difference champions are infinite square-free numbers containing any large primorial number as factor when x→∞.