Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime

formal-conjectures · oeis · ams-11

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.

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

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A078680
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/78680.lean
Local target: Corpus.OEIS78680.conjecture
Source SHA-256: 3a57c0e33786d8cf1513b945780d7868d5e6873207064d078645e89c17eafb7a
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record