FormalConjectures.OEIS.«282779»

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import FormalConjecturesUtil

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.

namespace OeisA282779open Nat Set

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

noncomputable def a (n : ℕ) : ℕ :=
  if n = 0 then 0 -- Handle the non-sequence index n=0
  else
    -- sInf computes the infimum of the set, which is the minimum since ℕ is well-ordered.
    sInf { T : ℕ | 0 < T ∧ ∀ k : ℕ, (k + T) ^ 3 % n = k ^ 3 % n }

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

noncomputable def periodOfPowerMod (p n : ℕ) : ℕ :=
  if n = 0 then 0
  else
    sInf { T : ℕ | 0 < T ∧ ∀ k : ℕ, (k + T) ^ p % n = k ^ p % n }

Conjecture: let $a_p(n)$ be the length of the period of the sequence $k^p \bmod n$ where $p$ is a prime, then $a_p(n) = n/p$ if $n \equiv 0 \pmod{p^2}$, else $a_p(n) = n$.

A formal proof has been found with the methods described in arxiv/2605.22763.

@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/282779.wip.lean#L104"]
theorem periodOfPowerMod_eq (p n : ℕ) (hp : Nat.Prime p) (hn : n > 0) :
    periodOfPowerMod p n = if p ^ 2 ∣ n then n / p else n := byp:ℕn:ℕhp:Nat.Prime phn:n > 0⊢ periodOfPowerMod p n = if p ^ 2 ∣ n then n / p else n
    sorryAll goals completed! 🐙

@[category test, AMS 11]
lemma a_1 : a 1 = 1 := by⊢ a 1 = 1 exact periodOfPowerMod_eq 3 1 (by⊢ Nat.Prime 3 decideAll goals completed! 🐙) (by⊢ 1 > 0 decideAll goals completed! 🐙)

@[category test, AMS 11]
lemma a_2 : a 2 = 2 := by⊢ a 2 = 2 exact periodOfPowerMod_eq 3 2 (by⊢ Nat.Prime 3 decideAll goals completed! 🐙) (by⊢ 2 > 0 decideAll goals completed! 🐙)

@[category test, AMS 11]
lemma a_3 : a 3 = 3 := by⊢ a 3 = 3 exact periodOfPowerMod_eq 3 3 (by⊢ Nat.Prime 3 decideAll goals completed! 🐙) (by⊢ 3 > 0 decideAll goals completed! 🐙)

@[category test, AMS 11]
lemma a_4 : a 4 = 4 := by⊢ a 4 = 4 exact periodOfPowerMod_eq 3 4 (by⊢ Nat.Prime 3 decideAll goals completed! 🐙) (by⊢ 4 > 0 decideAll goals completed! 🐙)

@[category test, AMS 11]
lemma a_5 : a 5 = 5 := by⊢ a 5 = 5 exact periodOfPowerMod_eq 3 5 (by⊢ Nat.Prime 3 decideAll goals completed! 🐙) (by⊢ 5 > 0 decideAll goals completed! 🐙)

end OeisA282779