Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime

formal-conjectures · oeis · ams-11

$a(n) = \min \{k \ge 1 \mid \text{Prime}(2 \cdot n^k - 1)\}$ for $n \ge 2$.

Is $a(n)$ defined for all $n \ge 2$?
That is, does there exist $k > 0$ such that $2 \cdot n^k - 1$ is prime?

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/A119591
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/119591.lean
Local target: Corpus.OEIS119591.conjecture
Source SHA-256: b47a159ff32fd2020e8904e4a5cc33efb3a7f7b8c4709d41fb5ee1bb02239f1a
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