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Claim. M. Ziller, On differences between consecutive numbers coprime to primorials, arXiv:2007.01808 (v1, 3 July 2020; no journal record). Write $F(k)$ for the least even integer that is not a difference $a_{i+1}-a_i$ of consecutive integers coprime to the $k$th primorial, as in Problem 854. Proposition 2.6 shows that $2k$ is such a difference for every $k$, by an explicit restricted covering of $1,\ldots,2k-1$ with one residue class modulo each of the first $k$ primes, and Proposition 2.7 shows that a difference for the $k$th primorial remains one for the $(k+1)$th. Together they give Corollary 2.8: each of $2,4,\ldots,2k$ occurs as a difference, so $F(k)\ge2k+2$. Ziller works with the periodic set of differences, which agrees with the site's finite one for $k\ge3$, since the boundary gap $2$ then also occurs inside $[1,n_k-1]$.

Section 4 reports an exhaustive computation, for every $k\le44$, of the least missing difference and of all missing differences below Jacobsthal's value $h(k)$, the largest gap. From these data it states Conjecture 4.1: for $k>1$ every even number up to $h(k-1)$ occurs as a difference for the $k$th primorial.

Covers. $F(k)\ge2k+2$ for every $k$ (for $k\ge3$ in the site's convention). Not covered: any bound of larger order, and the displayed question whether $\gg\max_i(a_{i+1}-a_i)$ even integers occur as gaps.

Standing. Claimed: an arXiv preprint with no journal record and no recorded reviewer. The site's label for the problem is OPEN, and the site does not mention the preprint.

Depends on. No page of this wiki.