[#P3042] Powerful Numbers in Products of Consecutive Integers

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Problem. Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?

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Definitions and notation

1Context

This problem is attributed to Paul Erdős and concerns a strengthening of questions about the arithmetic structure of products of consecutive integers. Erdős and Selfridge proved in 1975 that the product of $k \geq 2$ consecutive integers is never a perfect power. The present problem asks whether an even stronger restriction holds: that such a product always has at least one prime factor appearing to only the first power.

2Definitions

Definition 1 (A positive integer $N$). A positive integer $N$ is powerful if for every prime $p$ such that $p \mid N$, we also have $p^2 \mid N$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?^[1] Relevance to this problem For erdos problem 137, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number?^[1] Relevance to this problem For Powerful Numbers in Products of Consecutive Integers, record erdos-problem-137-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\cdots(n+k)$ is never a powerful number? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 137 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 137 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemPowerful Numbers in Products of Consecutive Integers

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Powerful Numbers in Products of Consecutive Integers.” TheoremDB. P3042. Problem statement; statement text SHA-256 6f9fa358ceca597660caefd4c6364f394d3c5579d836184fcc51e99932326e00. https://theoremdb.org/statement/?ref=P3042

BibTeX

@misc{theoremdb-problem-6f9fa358ceca597660caefd4c6364f394d3c5579d836184fcc51e99932326e00,
  title = {{Powerful Numbers in Products of Consecutive Integers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 6f9fa358ceca597660caefd4c6364f394d3c5579d836184fcc51e99932326e00},
  url = {https://theoremdb.org/statement/?ref=P3042}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 137, maintained status record. Erdős Problems record 137, checked 2026-08-01. Problem 137; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 137; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Powerful Numbers in Products of Consecutive Integers: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 137 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Powerful Numbers in Products of Consecutive Integers: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 137. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/137.lean:L33; theorem erdos_137; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Powerful Numbers in Products of Consecutive Integers: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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[#P3042] Powerful Numbers in Products of Consecutive Integers

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