Least k such that cyclotomic polynomial Φ_k(n) is prime

Is a(n) defined for all n ≥ 1? That is, for every n ≥ 1, does there exist k > 0 such that |Φ_k(n)| is prime?

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-117545,
  title        = {Least k such that cyclotomic polynomial Φ_k(n) is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-117545}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is $a (n)$ defined for all $n \geq 1$ ? That is, for every $n \geq 1$ , does there exist $k > 0$ such that $∣ Φ_{k} (n) ∣$ is prime?

$a (n) = min {k \in N ∣ 0 < k \land Prime (∣ Φ_{k} (n) ∣)}$ , where $Φ_{k} (n)$ is the $k$ -th cyclotomic polynomial evaluated at $n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«117545».

theorem conjecture (n : ℕ) (hn : 0 < n) :
    ∃ k > 0, ((Polynomial.cyclotomic k ℤ).eval (n : ℤ)).natAbs.Prime

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

A117545

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.