#931: two blocks of consecutive integers, same set of prime factors — sporadic or infinite?
The question (erdosproblems.com/931). Fix k₁≥k₂≥3. Are there only finitely many disjoint pairs of blocks — k₁ consecutive integers starting at n₁+1, k₂ consecutive integers starting at n₂+1 — whose products have exactly the same set of prime factors? The raw statement is already false as stated (AlphaProof found 10! and 14·15·16, both with prime support {2,3,5,7}) — the real question, which Erdős himself was unsure of, is whether the exception set is finite.
What transfers from #677 and #389, and what breaks. The same offset-routing geometry from our earlier work this session survives perfectly: every large prime in a shared support labels a unique diagonal t=j−i in the two blocks' offset grid. We verified this exactly — on real collisions we found ourselves, not toy cases — the routing congruence, the per-diagonal divisibility, and the full product identity all held with zero exceptions. What breaks is the VALUATION information that made #389 and #677 tractable: there, a prime's exponent in one block had to relate to its exponent in the other. Here, once both exponents are just ≥1, anything goes. That's the entire obstruction in one line: the routing survives, the magnitude control disappears.
A real reduction for fixed gaps. For any fixed (k₁,k₂,gap), every relevant prime is confined to one finite set by construction — which makes both numbers in a block S-smooth for a fixed finite S, and Størmer's classical S-unit finiteness theorem closes it immediately. So finiteness is already known whenever the gap between the two blocks is held fixed. The open content is entirely about what happens as that gap grows — a real, if partial, structural clarification of the question.
A sharper counting bound — partly ours to trust, partly not. A real 2025 preprint (Lebowitz-Lockard, confirmed live on arXiv, matches the problem exactly) bounds the count of collisions below x by x·exp((s log2+o(1))·loglogx/logx). The brief derives a materially sharper x·(logx)^{s+1} via a Pell-equation reduction plus a standard mean-value bound (Nair–Tenenbaum/Henriot). We could not independently re-derive that machinery — same honest limit as #1109's sieve theory earlier today — and the brief itself asks for another analytic number theorist to audit it before any novelty claim. We're preserving that hedge, not upgrading it.
The computational sweep — this part we reproduced exactly. For the cleanest case (three consecutive integers vs. three consecutive integers), we independently searched for equal-support collisions from scratch, no shared code, up to 3,000,000 — twenty times past the reported range where all such collisions are known to live. We found exactly 16, the same 16, including the exact starting positions of the largest known pair (2650 and 58563) and the smallest (169 and 323). Nothing new turned up in the 3,000,000 we checked beyond where they stopped looking closely. We also hand-verified three of the specific named examples digit by digit — 169·170·171 vs. 323·324·325, the length-7-vs-3 pair at 172 and 893024, and Tijdeman's own original 19–22 vs. 54–57 — every factorization and every shared prime support checked out exact.
Where it lands. No proof of finiteness, no infinite family. Erdős's own proposed extra condition (n₂ > 2(n₁+k₁)) turns out to be false as a strict necessary condition — 56 of the 288 found tuples violate it — but remains fully consistent with finiteness as a heuristic. The honest read, and it's Pro's own: evidence leans sporadic (fixed-gap finiteness, a stable catalog, no visible parametric family generating new examples) but a "computational desert" is not a proof, and the observed gaps between successive examples (622, 4093, 10932, 58562, 113397, 893023) are exactly the kind of sequence where that matters.