Smallest even value missing from the first $x$ prime gaps: does $r(x)\to\infty$? (Erdős #853)
Statement
Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solution with $n\le x$ — that is, the least even value not yet appearing among the first $x$ prime gaps. Is it true that $r(x)\to\infty$? Or even that $r(x)/\log x\to\infty$?
Acceptance. FULLY RESOLVES: a rigorous proof that $r(x)\to\infty$ (or that it does not), and/or of the stronger $r(x)/\log x\to\infty$ (or its refutation), with complete justification. ADVANCES: prove the weaker claim $r(x)\to\infty$ unconditionally or under a precisely stated hypothesis; or establish any nontrivial lower bound $r(x)\ge g(x)$ with $g(x)\to\infty$; or computationally extend the tabulation of $r(x)$ to a record $x$, supplying the enumeration program, the value of $r(x)$ and the smallest missing even gap at that $x$, and the extended OEIS A390769 data. Deliver the proof or the computation code with the extended sequence.
Background
Posed by Erdős [Er85c]; listed as open on erdosproblems.com/853 (fetched 2026-07-21, status 'open'), with a machine-formalised statement in the Google DeepMind formal-conjectures project. As Bloom notes, Erdős stated the problem without the evenness restriction on $t$, but it is clearly necessary: apart from $d_1=p_2-p_1=1$ all prime gaps are even, so an odd $t$ trivially never occurs. The question is tied to Polignac's conjecture (that every even number arises infinitely often as a prime gap): if every even value eventually appears, $r(x)$ still measures how slowly the last-appearing small even gaps fill in. The gaps are OEIS A001223, and the least-missing-even-gap function is tracked in OEIS A390769. Attacker's tool: enumeration — compute the prime gaps $d_n$ for $n\le x$, record which even values have occurred, take the least even value still missing to get $r(x)$, extend OEIS A390769, and test both growth conjectures against the tabulated data.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #853 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #853 (DeepMind formal-conjectures) | website |
| REF-03 | OEIS A001223 — differences between consecutive primes | website |
| REF-04 | OEIS A390769 — least even number not yet a prime gap up to x | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.