Erdős Problem #1209 — open problem in number theory

Erdős Problem #1209

Are there n such that n+2^2^k is always squarefree?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-1209,
  title        = {Erdős Problem #1209},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1209}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_1209.parts.iii.b. Are there $n$ such that $n + 2^{2^{k}}$ is always squarefree?

erdos_1209.parts.iii.c. Are there $n$ such that $n + 2^{2^{k}}$ is infinitely often a prime?

erdos_1209.parts.iii.d. Are there $n$ such that $n + 2^{2^{k}}$ is infinitely often squarefree?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1209» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1209.parts.iii.b :
    answer(sorry) ↔ ∃ n : ℕ, ∀ k : ℕ, Squarefree (n + 2 ^ (2 ^ k))

theorem erdos_1209.parts.iii.c :
    answer(sorry) ↔ ∃ n : ℕ, {k | (n + 2 ^ (2 ^ k)).Prime}.Infinite

theorem erdos_1209.parts.iii.d :
    answer(sorry) ↔ ∃ n : ℕ, {k | Squarefree (n + 2 ^ (2 ^ k))}.Infinite

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1209. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/429
erdosproblems.com/1102
erdosproblems.com/1209
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.