Smallest m > 0 such that n · 2^m + 1 is prime

There is a conjecture that the first zero is n = 65536 = 2^16 (which is equivalent to the statement that 2^2^k + 1 is composite for k > 4). - _T. D. Noe_, Feb 25 2011

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-78680,
  title        = {Smallest m > 0 such that n · 2^m + 1 is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-78680}},
  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

There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to the statement that $2^{2^{k}} + 1$ is composite for $k > 4$ ). - _T. D. Noe_, Feb 25 2011

The sequence $a (n)$ is the smallest positive integer $m$ such that $n \cdot 2^{m} + 1$ is prime, or $0$ if no such $m$ exists.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«78680».

theorem conjecture :
    a (2 ^ 16) = 0 ∧ ∀ n : ℕ, 1 ≤ n ∧ n < 2 ^ 16 → a n ≠ 0

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

A078680

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.