Can a product of k ≥ 3 consecutive integers ever be powerful? (Erdős #137)
Statement
An integer $N$ is 'powerful' if $p\mid N$ implies $p^2\mid N$ for every prime $p$. Fix an integer $k\ge 3$. Can the product $n(n+1)\cdots(n+k-1)$ of $k$ consecutive positive integers ever be powerful? Erdős and Selfridge conjecture that the answer is no: for every $k\ge 3$, no product of $k$ consecutive positive integers is powerful.
Acceptance. FULLY RESOLVES: either (a) EXHIBIT an integer $k\ge 3$ and a positive integer $n$ such that $n(n+1)\cdots(n+k-1)$ is powerful — giving the prime factorisation of the product and certifying every prime occurs to exponent $\ge 2$, a fully machine-checkable witness that disproves the conjecture; or (b) PROVE the Erdős–Selfridge conjecture that no such product is powerful (Lean/Coq formalisation preferred, else a complete written proof), possibly conditional on a clearly-flagged hypothesis such as abc. ADVANCES (each checkable): establish and document a reproducible verified range — prove by exhaustive search that no product of $k$ consecutive integers (for $k$ in a stated range and $n$ up to a stated bound $N$) is powerful, delivering the search code and an exhaustiveness certificate (no explicit computational bound is recorded in the sources cited here, so any documented, reproducible range is progress); or prove the weaker [Er82c] conjecture (at least $k$ primes dividing the product to the first power) in new cases. Deliver the explicit powerful witness plus factorisation, or the proof file, or the search program plus attained bound and certificate.
Background
Conjectured by Erdős and Selfridge; recorded in Erdős–Graham [ErGr80] and repeated in [Er82c, p.28] and [Er97c]. Listed as open on erdosproblems.com/137 (fetched 2026-07-13, status 'open', tagged 'number theory | powerful'); a Lean 4 formalisation exists in google-deepmind/formal-conjectures. Context: for $k=2$ the answer is YES — there are infinitely many $n$ with $n(n+1)$ powerful (Erdős #364, erdosproblems.com/364), coming from Pell solutions. For $k\ge 3$ no example is known and none is expected. The strongest related theorem is Erdős–Selfridge [ErSe75]: a product of $k\ge 2$ consecutive integers is never a PERFECT POWER — but 'powerful' is far weaker than 'perfect power', and Erdős called the powerful version 'hopeless at present'. In [Er82c] Erdős conjectured a stronger statement: for fixed $k$ and all sufficiently large $n$, every product $m(m+1)\cdots(m+n)$ has at least $k$ distinct primes $p$ with $p\mid m(m+1)\cdots(m+n)$ yet $p^2\nmid m(m+1)\cdots(m+n)$. Closely related venue neighbours: three consecutive powerful numbers (Erdős #364) and consecutive $2$-full/$3$-full pairs (Erdős #366, erdosproblems.com/366). No Erdős prize is recorded. Attacker's tool: a factorisation-based search — enumerate windows of $k$ consecutive integers and test their product for powerfulness (strip the squarefull part and check every remaining prime has exponent $\ge 2$), documenting a verified no-example range; on the theory side, abc-style or $S$-unit bounds on the primes dividing such products.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #137 (T. F. Bloom) | website |
| REF-02 | Lean 4 formalisation — google-deepmind/formal-conjectures #137 | website |
Investigations · 0
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