282779.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

8 declarations, 1 contain sorry, 0 declared as axioms

Period of $p$-th powers modulo $n$

a: Period of cubes mod $n$. The $n$-th term $a(n)$ is the smallest positive integer $T$ such that $\forall k \in \mathbb{N}$, $(k+T)^3 \equiv k^3 \pmod n$.

The length of the minimal positive period of the sequence $k^p \pmod n$. $a_p(n) = \min \{ T \in \mathbb{N}^+ \mid \forall k \in \mathbb{N}, (k+T)^p \equiv k^p \pmod n \}$.

References:

A282779
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA282779.a Period of cubes mod $n$ line 47 source
Source documentation
Period of cubes mod $n$. The $n$-th term $a(n)$ is the smallest positive integer $T$ such that $\forall k \in \mathbb{N}$, $(k+T)^3 \equiv k^3 \pmod n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, InfSet.sInf, Set.ofPred, And, LT.lt, HMod.hMod, HPow.hPow, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def OeisA282779.periodOfPowerMod The length of the minimal positive period of the sequence $k^p \pmod n$ line 57 source
Source documentation
The length of the minimal positive period of the sequence $k^p \pmod n$. $a_p(n) = \min \{ T \in \mathbb{N}^+ \mid \forall k \in \mathbb{N}, (k+T)^p \equiv k^p \pmod n \}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, InfSet.sInf, Set.ofPred, And, LT.lt, HMod.hMod, HPow.hPow, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
noncomputable def periodOfPowerMod (p n : ℕ) : ℕ :=
…
```
theorem OeisA282779.periodOfPowerMod_eq sorry line 72 source
depends on sorry what this means
Statement uses: Nat, Nat.Prime, GT.gt, OfNat.ofNat, Eq, OeisA282779.periodOfPowerMod, ite, Dvd.dvd, HPow.hPow, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA282779.a_1 line 78 source
depends on sorry what this means
Statement uses: Eq, Nat, OeisA282779.a, OfNat.ofNat
Proof uses: OeisA282779.periodOfPowerMod_eq, of_decide_eq_true, Nat.Prime, id, Bool, Decidable.decide, Bool.true, Eq.refl, GT.gt
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA282779.a_2 line 81 source
depends on sorry what this means
Statement uses: Eq, Nat, OeisA282779.a, OfNat.ofNat
Proof uses: OeisA282779.periodOfPowerMod_eq, of_decide_eq_true, Nat.Prime, id, Bool, Decidable.decide, Bool.true, Eq.refl, GT.gt
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA282779.a_3 line 84 source
depends on sorry what this means
Statement uses: Eq, Nat, OeisA282779.a, OfNat.ofNat
Proof uses: OeisA282779.periodOfPowerMod_eq, of_decide_eq_true, Nat.Prime, id, Bool, Decidable.decide, Bool.true, Eq.refl, GT.gt
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA282779.a_4 line 87 source
depends on sorry what this means
Statement uses: Eq, Nat, OeisA282779.a, OfNat.ofNat
Proof uses: OeisA282779.periodOfPowerMod_eq, of_decide_eq_true, Nat.Prime, id, Bool, Decidable.decide, Bool.true, Eq.refl, GT.gt
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA282779.a_5 line 90 source
depends on sorry what this means
Statement uses: Eq, Nat, OeisA282779.a, OfNat.ofNat
Proof uses: OeisA282779.periodOfPowerMod_eq, of_decide_eq_true, Nat.Prime, id, Bool, Decidable.decide, Bool.true, Eq.refl, GT.gt
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.