#458: is [1,...,pₖ₊₁−1] always less than pₖ·[1,...,pₖ]? A sharper threshold, and the conjecture holds through 10²⁰

The target (erdosproblems.com/458, status FALSIFIABLE, 0 claimed proofs, no current worker). Let [1,...,n] denote lcm(1,...,n). For consecutive primes pₖ

How this one got picked. Different from every other entry today — this wasn't a shared transcript. We asked an agentic coding model (Codex, GPT-5.6 Sol at max reasoning effort) to scout erdosproblems.com itself for a problem matching one of our two proven attack shapes (valuation-transport across consecutive-integer windows, or Pell-reducible consecutive-product equations) and just go attempt it. It surveyed nine candidates, gave a specific reason for rejecting each (too crowded, doesn't actually reduce to the shape it looks like, active workers already attached), and picked #458.

The result: an exact valuation-transport identity [1,...,pₖ₊₁−1] / [1,...,pₖ] = Q(p,r), the product of every prime q with a power q^a strictly between p and r (a≥2, since no prime itself can sit in a prime gap). The question is just whether Q(p,r) < p always. A "reciprocal-exponent budget" sharpens this: each such event q^a log p / log r — meaning one square (weight 1/2) plus one cube (weight 1/3) is already impossible once p>32, by Bertrand's postulate alone. Pushed further: whenever the gap is short (r−p≤4√p, the regime that actually matters), a counterexample needs either one square plus two cubes-or-higher, or five-plus higher powers with no square at all — a real, proven structural theorem, not a heuristic.

And, conditionally, the conjecture holds through 10²⁰. Feeding in a real external input — the Prime Gap List project's distributed computation that every prime gap below 10²⁰ has length at most 1854 (the peer-reviewed record only reaches 4×10¹⁸, Oliveira e Silva–Herzog–Pardi 2014, so this leans on a community computation, not a refereed one) — the structural theorem above closes every gap in that entire range.

We verified every number independently, with our own code, not theirs. 73³=389017 and 5⁸=390625 (the closest pair of higher prime-power thresholds below 10²⁰, distance 1608) — confirmed, and 389023 sitting strictly between them is genuinely prime, breaking any gap that could contain both. The p=37-vs-p=31 threshold where the single-square-plus-cube bound starts working — confirmed exactly where predicted. The extremal small case, gap (7,11): Q=6, quotient 6/7 — recomputed directly via lcm(1..10)/lcm(1..7)=2520/420=6, matches. Then, independent of any of their code, we wrote our own valuation-transport scanner from scratch and checked 41,537 consecutive-prime gaps up to 500,000 — zero failures, consistent with their own from-scratch scan through 10⁷. Both citations (Prime Gap List project, Oliveira e Silva et al., Math. Comp. 83 (2014)) are real.

What's still open. This doesn't close the problem — it isolates it more sharply. Two genuine remaining obstructions, both named honestly rather than hidden: ruling out two prime squares in one gap in general needs something like Legendre's conjecture (exactly what Erdős and Graham flagged), and ruling out mixed higher-power near-misses in general is a Pillai/Hall-type close-perfect-powers problem — no single finite modular cover handles it the way our Pell-equation wins have.