The $k$-Tuple Jumping Champions among Consecutive Primes

For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to $x$. For $k=1$, it's known as the jumping champion introduced by J. H. Conway. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf announced the Jumping Champion Conjecture that the jumping champions greater than 1 are 4 and the primorials 2, 6, 30, 210, 2310,.... They also made a weaker and possibly more accessible conjecture that any fixed prime $p$ divides all sufficiently large jumping champions. These two conjectures were proved by Goldston and Ledoan under the assumption of appropriate forms of the Hardy-Littlewood conjecture recently. In the present paper we consider the situation for any $k\ge2$ and prove that any fixed prime $p$ divides every element of all sufficiently large $k$-tuple jumping champions under the assumption that the Hardy-Littlewood prime $k+1$-tuple conjecture holds uniformly for $\mathcal{D}_k\subset[2,\log^{k+1}x]$. With a stronger form of the Hardy-Littlewood conjecture, we also proved that, for any sufficiently large $k$-tuple jumping champion, the $gcd$ of elements in it is square-free.