117545.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 1 contain sorry, 0 declared as axioms

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

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

References:

A117545

Imports (1)

FormalConjecturesUtil

def OeisA117545.a Least $k > 0$ such that $|\Phi_k(n)|$ is prime, or $0$ if no such $k$ exists line 34 source
Source documentation
Least $k > 0$ such that $|\Phi_k(n)|$ is prime, or $0$ if no such $k$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic
7 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA117545.a_1 Value of the sequence a at 1 line 39 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117545.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Polynomial and 44 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by
…
```
Mentions: OeisA117545.a
theorem OeisA117545.a_2 Value of the sequence a at 2 line 62 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117545.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Polynomial and 52 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by
…
```
Mentions: OeisA117545.a
theorem OeisA117545.a_3 Value of the sequence a at 3 line 85 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117545.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Polynomial and 27 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA117545.a
theorem OeisA117545.a_4 Value of the sequence a at 4 line 100 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117545.a, OfNat.ofNat
Proof uses: IsLeast, Set.ofPred, And, LT.lt, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, And.intro, Membership.mem, Set, lowerBounds, id, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Polynomial and 29 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by
…
```
Mentions: OeisA117545.a
theorem OeisA117545.conjecture Is $a(n)$ defined for all $n \ge 1$? sorry line 117 source
Source documentation
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?

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, And, GT.gt, Nat.Prime, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 0 < n) :
    ∃ k > 0, ((Polynomial.cyclotomic k ℤ).eval (n : ℤ)).natAbs.Prime := by
…
```