FormalConjectures.OEIS.«117545»

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import FormalConjecturesUtil

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

namespace OeisA117545

Least $k > 0$ such that $|\Phi_k(n)|$ is prime, or $0$ if no such $k$ exists.

noncomputable def a (n : ℕ) : ℕ :=
  sInf {k : ℕ | 0 < k ∧ ((Polynomial.cyclotomic k ℤ).eval (n : ℤ)).natAbs.Prime}

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 2 := by⊢ a 1 = 2
  have h_least : IsLeast {k : ℕ | 0 < k ∧ ((Polynomial.cyclotomic k ℤ).eval (1 :
  ℤ)).natAbs.Prime} 2 := by
    constructorleft⊢ 2 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs}right⊢ 2 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
    ·left⊢ 2 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2 simp only [Set.mem_ofPred_eq]left⊢ 0 < 2 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 2 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      refine ⟨by⊢ 0 < 2h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2 decideAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2, ?_⟩
      have : Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1 := Polynomial.cyclotomic_two ℤleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 2 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      rw [thisleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 1 (Polynomial.X + 1)).natAbsleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 1 (Polynomial.X + 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2]leftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 1 (Polynomial.X + 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      norm_numAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
    ·right⊢ 2 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2 intro k hkrightk:ℕhk:k ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs}⊢ 2 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      simp only [Set.mem_ofPred_eq] at hkrightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs⊢ 2 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      by_contra! hrightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbsh:k < 2⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      have hk_pos := hk.1rightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbsh:k < 2hk_pos:0 < k⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      interval_cases kright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      have hk2 := hk.2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbs⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      have h1 : Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1 := Polynomial.cyclotomic_one ℤright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      rw [h1right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 1 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 1 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2] at hk2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 1 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      revert hk2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1h1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 1 (Polynomial.X - 1)).natAbs → Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
      norm_numh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 1 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 1 = 2
  exact h_least.csInf_eqAll goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 2 := by⊢ a 2 = 2
  have h_least : IsLeast {k : ℕ | 0 < k ∧ ((Polynomial.cyclotomic k ℤ).eval (2 :
  ℤ)).natAbs.Prime} 2 := by
    constructorleft⊢ 2 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs}right⊢ 2 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
    ·left⊢ 2 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2 simp only [Set.mem_ofPred_eq]left⊢ 0 < 2 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 2 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      refine ⟨by⊢ 0 < 2h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2 decideAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2, ?_⟩
      have : Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1 := Polynomial.cyclotomic_two ℤleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 2 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      rw [thisleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 2 (Polynomial.X + 1)).natAbsleftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 2 (Polynomial.X + 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2]leftthis:Polynomial.cyclotomic 2 ℤ = Polynomial.X + 1⊢ Nat.Prime (Polynomial.eval 2 (Polynomial.X + 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      norm_numAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
    ·right⊢ 2 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2 intro k hkrightk:ℕhk:k ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs}⊢ 2 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      simp only [Set.mem_ofPred_eq] at hkrightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs⊢ 2 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      by_contra! hrightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbsh:k < 2⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      have hk_pos := hk.1rightk:ℕhk:0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbsh:k < 2hk_pos:0 < k⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      interval_cases kright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      have hk2 := hk.2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbs⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      have h1 : Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1 := Polynomial.cyclotomic_one ℤright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      rw [h1right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 2 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseright.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 2 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2] at hk2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1hk2:Nat.Prime (Polynomial.eval 2 (Polynomial.X - 1)).natAbsh1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      revert hk2right.«1»k:ℕhk:0 < 1 ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic 1 ℤ)).natAbsh:1 < 2hk_pos:0 < 1h1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 2 (Polynomial.X - 1)).natAbs → Falseh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
      norm_numh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 2 (Polynomial.cyclotomic k ℤ)).natAbs} 2⊢ a 2 = 2
  exact h_least.csInf_eqAll goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 1 := by⊢ a 3 = 1
  have h_least : IsLeast {k : ℕ | 0 < k ∧ ((Polynomial.cyclotomic k ℤ).eval (3 :
  ℤ)).natAbs.Prime} 1 := by
    constructorleft⊢ 1 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs}right⊢ 1 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
    ·left⊢ 1 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1 simp only [Set.mem_ofPred_eq]left⊢ 0 < 1 ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic 1 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
      refine ⟨by⊢ 0 < 1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1 decideAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1, ?_⟩
      have h1 : Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1 := Polynomial.cyclotomic_one ℤlefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic 1 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
      rw [h1lefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 3 (Polynomial.X - 1)).natAbslefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 3 (Polynomial.X - 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1]lefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 3 (Polynomial.X - 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
      norm_numAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
    ·right⊢ 1 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1 intro k hkrightk:ℕhk:k ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs}⊢ 1 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
      exact hk.1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 3 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 3 = 1
  exact h_least.csInf_eqAll goals completed! 🐙

Value of the sequence a at 4.


@[category test, AMS 11]
theorem a_4 : a 4 = 1 := by⊢ a 4 = 1
  have h_least : IsLeast {k : ℕ | 0 < k ∧ ((Polynomial.cyclotomic k ℤ).eval (4 :
  ℤ)).natAbs.Prime} 1 := by
    constructorleft⊢ 1 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs}right⊢ 1 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
    ·left⊢ 1 ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1 simp only [Set.mem_ofPred_eq]left⊢ 0 < 1 ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic 1 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
      refine ⟨by⊢ 0 < 1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1 decideAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1, ?_⟩
      have h1 : Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1 := Polynomial.cyclotomic_one ℤlefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic 1 ℤ)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
      rw [h1lefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 4 (Polynomial.X - 1)).natAbslefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 4 (Polynomial.X - 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1]lefth1:Polynomial.cyclotomic 1 ℤ = Polynomial.X - 1⊢ Nat.Prime (Polynomial.eval 4 (Polynomial.X - 1)).natAbsh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
      norm_numAll goals completed! 🐙h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
    ·right⊢ 1 ∈ lowerBounds {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs}h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1 intro k hkrightk:ℕhk:k ∈ {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs}⊢ 1 ≤ kh_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
      exact hk.1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1h_least:IsLeast {k | 0 < k ∧ Nat.Prime (Polynomial.eval 4 (Polynomial.cyclotomic k ℤ)).natAbs} 1⊢ a 4 = 1
  exact h_least.csInf_eqAll goals completed! 🐙

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?

@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) :
    ∃ k > 0, ((Polynomial.cyclotomic k ℤ).eval (n : ℤ)).natAbs.Prime := byn:ℕhn:0 < n⊢ ∃ k > 0, Nat.Prime (Polynomial.eval (↑n) (Polynomial.cyclotomic k ℤ)).natAbs
  sorryAll goals completed! 🐙

end OeisA117545