Least $k$ such that cyclotomic polynomial $\Phi_k(n)$ is prime

formal-conjectures · oeis · ams-11

$a(n) = \min \{k \in \mathbb{N} \mid 0 < k \wedge \text{Prime}(|\Phi_k(n)|) \}$,
where $\Phi_k(n)$ is the $k$-th cyclotomic polynomial evaluated at $n$.

Is $a(n)$ defined for all $n \ge 1$?
That is, for every $n \ge 1$, does there exist $k > 0$ such that $|\Phi_k(n)|$ 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/A117545
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/117545.lean
Local target: Corpus.OEIS117545.conjecture
Source SHA-256: f0b9e3fdda73b9164d898555c71d89720e5cf0c9b2e4b88ab04c2de3ea3ea778
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