a(n) = 2^(2^n) — open problem in number theory

a(n) = 2^(2^n)

I conjecture that a(n) ; n>1 are the numbers such that n^4-1 divides 2^n-1, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-1146,
  title        = {a(n) = 2^(2^n)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-1146}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

I conjecture that { $a (n)$ ; $n > 1$ } are the numbers such that $n^{4} - 1$ divides $2^{n} - 1$ , intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«1146».

theorem conjecture :
  ∀ k : ℕ, ((k^4 - 1) : ℕ) ∣ (2^k - 1 : ℕ) → k > 1 → ∃ n : ℕ, 2 ≤ n ∧ k = a n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A001146

Source and licence

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