Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$

formal-conjectures · oeis · ams-11

Define $F(n) = 2^{2^n} + 1$ (the $n$-th Fermat number) and $M(n) = 2^n - 1$ (the $n$-th Mersenne
number). Then $a(n) = F(n) + M(n) - 1 = 2^{2^n} + 2^n - 1$.

The first 5 entries are primes. Are there infinitely many primes in this sequence?

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/A119563
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/119563.lean
Local target: Corpus.OEIS119563.conjecture
Source SHA-256: 4c3ea02993137cc4df0d7557f20b29912d73181b74ff64a15e4673636fcaee0e
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