Least prime missing from a product of $k$ consecutive integers: is $q(n,k)<(1+o(1))\log n$? (Erdős #663)
Statement
Let $k\geq 2$ be fixed, and for $n\geq 1$ let $q(n,k)$ denote the least prime that does not divide the product $\prod_{1\leq i\leq k}(n+i)$ of $k$ consecutive integers. Is it true that, for each fixed $k$ and all sufficiently large $n$, $$q(n,k)<(1+o(1))\log n?$$
Acceptance. FULLY RESOLVES: for each fixed $k\geq 2$, prove that $q(n,k)<(1+o(1))\log n$ for all sufficiently large $n$, or disprove it (a proof that $\limsup_n q(n,k)/\log n>1$ for some fixed $k$); a complete proof, Lean/Coq-checkable preferred. ADVANCES: prove a bound of the form $q(n,k)<(c+o(1))\log n$ with a constant $c$ strictly smaller than the easy value $k$ for a given $k$, or establish the conjectured $(1+o(1))\log n$ bound in a growing range such as $k=o(\log n)$, with proof; alternatively deliver an extensive, reproducible computation of $q(n,k)$ over a certified range documenting the empirical constant and extending OEIS A391668. Deliver the proof or the search code plus certified data. State in words the bound you improve on.
Background
A problem of Erdős and Pomerance, recorded in [BEGL96] and [Er97e]; listed as open on erdosproblems.com/663 (fetched 2026-07-21, status 'open'). The weaker bound $q(n,k)<(1+o(1))k\log n$ is easy: the primes up to $(1+o(1))k\log n$ cannot all divide a product of $k$ consecutive integers of size about $n^k$. The conjecture is that the factor $k$ can be removed for each fixed $k$, leaving $(1+o(1))\log n$; Erdős even asks whether the improved bound persists as $k$ grows, perhaps up to $k=o(\log n)$. Terence Tao has supplied a supporting heuristic in the site comments, but no proof is known. The problem is a cousin of Erdős #457 (erdosproblems.com/457), and OEIS A391668 tabulates related values. The attacker's tool: analytic/sieve number theory controlling which small primes are forced to divide a block of consecutive integers, backed by large-scale computation of $q(n,k)$ to test the constant, probe the $k=o(\log n)$ range, and extend OEIS A391668.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #663 (T. F. Bloom) | website |
| REF-02 | OEIS A391668 — related to the least prime not dividing a product of consecutive integers | website |
Investigations · 0
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