# P3042: Powerful Numbers in Products of Consecutive Integers

- ID: `P3042`
- Reference: `erdos-problem-137`
- Page: https://theoremdb.org/statements/P3042
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

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

### What 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?

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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?

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?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-137`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 137, maintained status record. Erdős Problems record 137, checked 2026-08-01. Problem 137; status field and linked bibliography https://www.erdosproblems.com/137
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 137 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 137. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/137.lean:L33; theorem erdos_137; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/137.lean#L33
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.