# P2964: Erdős's squarefree-power-of-two sum problem

- ID: `P2964`
- Reference: `erdos-problem-11`
- Page: https://theoremdb.org/statements/P2964
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$?

### Context

This is a classical additive number theory problem posed by Paul Erdős. Related variants have been studied, including whether every integer $n > 1$ not divisible by $4$ has this property, and whether every odd $n > 1$ can be written as the sum of a squarefree number and two powers of $2$. The statement has been verified computationally for all odd $n$ with $1 < n < 2^{50}$.

### Problem setup

- **Definition (A positive integer).** A positive integer is called squarefree if it is not divisible by the square of any prime.
- **Remark.** This is a classical additive number theory problem posed by Paul Erdős. Related variants have been studied, including whether every integer $n > 1$ not divisible by $4$ has this property, and whether every odd $n > 1$ can be written as the sum of a squarefree number and two powers of $2$. The statement has been verified computationally for all odd $n$ with $1 < n < 2^{50}$.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 11 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$? [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 11 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 11 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 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 11 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 11 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 11: A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-11`, 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 11, maintained status record. Erdős Problems record 11, checked 2026-08-01. Problem 11; status field and linked bibliography https://www.erdosproblems.com/11
   - Also cited at Problem 11; 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 Erdős's squarefree-power-of-two sum problem: 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 11 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 Erdős's squarefree-power-of-two sum problem: 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 11. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/11.lean:L31; theorem erdos_11; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/11.lean#L31
   - 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 Erdős's squarefree-power-of-two sum problem: Supplies the pinned formal declaration whose human-readable editorial statement is published here.