[#P2964] Erdős's squarefree-power-of-two sum problem
Contents
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 $2$?
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Definitions and notation
1Context
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 $1 < n < 2^{50}$.
2Definitions
Definition 1 (A positive integer). A positive integer is called squarefree if it is not divisible by the square of any prime.
3What 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 $2$?
1Status
Saved packet · July 31, 2026
Saved packet status (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 $2$?[1]
1Packet records
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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 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.
How the 2 records connect
ProblemErdős's squarefree-power-of-two sum problem
All 1 recorded relations between these records and the problem
2See also
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Cite this problem statement
Cite the original sources separately.
“Erdős's squarefree-power-of-two sum problem.” TheoremDB. P2964. Problem statement; statement text SHA-256 e7e543219d14d6e9c92e7b36adb5da933cf6f89e6b153db863ff92d5d51a542b. https://theoremdb.org/statement/?ref=P2964
@misc{theoremdb-problem-e7e543219d14d6e9c92e7b36adb5da933cf6f89e6b153db863ff92d5d51a542b,
title = {{Erdős's squarefree-power-of-two sum problem}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 e7e543219d14d6e9c92e7b36adb5da933cf6f89e6b153db863ff92d5d51a542b},
url = {https://theoremdb.org/statement/?ref=P2964}
}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 11, maintained status record. Erdős Problems record 11, checked 2026-08-01. Problem 11; 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 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.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.
- Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 11 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 Erdős's squarefree-power-of-two sum problem: Pins the maintained database snapshot used for this release's dated status decision.
- Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 11. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/11.lean:L31; theorem erdos_11; 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 Erdős's squarefree-power-of-two sum problem: 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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