Products of consecutive integers over disjoint long intervals: never a perfect power? (Erdős #930)
Statement
Is it true that, for every $r$, there is a $k$ such that whenever $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, the combined product $$\prod_{1\leq i\leq r}\prod_{m\in I_i}m$$ is not a perfect power (an integer of the form $t^s$ with $s\geq 2$)?
Acceptance. FULLY RESOLVES: a complete proof — machine-checkable (Lean/Coq) preferred, else a full written proof — that for every $r$ there is a threshold $k(r)$ with the stated property; OR a disproof exhibiting some fixed $r$ for which arbitrarily long disjoint $r$-interval products can be perfect powers, given as an explicit infinite family together with a proof that its members are perfect powers. ADVANCES: settle the first open case $r=2$ (prove existence of a suitable $k$, i.e. that sufficiently long disjoint pairs of intervals never multiply to a perfect power), or any single fixed $r$, with proof; or construct an explicit infinite family of perfect-power products arising from disjoint long intervals for some $r$ (a structural obstruction), with proof; or a machine-verified reduction of the general statement to a finite/effective check. Deliver the proof, or the infinite family together with its correctness proof.
Background
Posed by Erdős [Er76d]. The base case $r=1$ is the celebrated Erdős–Selfridge theorem [ErSe75] that a product of two or more consecutive integers is never a perfect power. The requirement that the intervals be long relative to $r$ is genuinely necessary: already for $r=2$ one can, without a lower bound on the interval lengths, construct disjoint pairs of intervals whose combined product is a perfect power — see the constructions collected under Erdős #363 (erdosproblems.com/363), which also treats the analogous question for squares. So the content of #930 is that once each interval is forced to be long enough (in terms of $r$), no such coincidence can occur. Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/930). Listed as open on erdosproblems.com/930 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. Attacker's tool: analytic number theory in the Erdős–Selfridge tradition (large prime factors of products of consecutive integers, abc-type inputs) toward the proof, backed by a computational falsification channel that hunts for disjoint long-interval products that are perfect powers.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #930 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #930 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.