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Claim. 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. The Erdős Problem-a-Day working report on Problem 854, of 27 July 2026, proves

$$\liminf_{k\to\infty}\frac{F(k)}{k\log\log k}\ge e^{\gamma},$$

with $\gamma$ Euler's constant, so $F(k)\ge(e^{\gamma}-o(1))\,k\log\log k$, and as a corollary that at least $(e^{\gamma}/2-o(1))\,k\log\log k$ distinct even gaps occur. The report also computes $F(k)$ exactly for $2\le k\le13$ (from $F(2)=6$ to $F(13)=76$) and certifies $F(100)\ge482$ by an explicit covering.

Covers. The lower bounds above for the least even non-gap and for the number of distinct even gaps. Not covered: an upper bound for $F(k)$, and the displayed question whether $\gg\max_i(a_{i+1}-a_i)$ even integers occur as gaps.

Standing. The claimant is Patrick White, who published the result on 2026-07-27 as a working report on erdosproblemaday.com, a public ledger of AI-assisted reports that names the report's authors as Patrick White and Claude (Anthropic) and labels this entry PARTIAL. The report is not on the site's proof-claims tab, is not refereed, and no reviewer is recorded, so the claim stays claimed. The site's label for the problem is OPEN.

Depends on. No page of this wiki.