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

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A finite mathematical diagram showing a squarefree integer and a power of two joined by addition.
A squarefree integer and a power of two joined by addition.

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 $2$?

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 $1 < n < 2^{50}$.

2Problem setup

Definition 1 (A positive integer). A positive integer is called squarefree if it is not divisible by the square of any prime.

Remark 1. 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 $1 < n < 2^{50}$.

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

Current 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

2 records

RecordKindAssessment
ResultSupported
TraceReported
Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow
How the records connect to the problem

ProblemErdős's squarefree-power-of-two sum problem

  1. Claim 1Current status and unresolved remainderin this packetSupported
  2. Route 1Resolve the stated acceptance conditionaddressesReported

2See also

How to cite

TheoremDB contributors, “Erdős's squarefree-power-of-two sum problem,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-11

This page as plain text: erdos-problem-11.md

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

1References

  1. 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.
  2. 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.
  3. 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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