FormalConjectures.OEIS.«70518»

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import FormalConjecturesUtil

Value of $n$-th cyclotomic polynomial at $n$

The sequence $a(n) = \Phi_n(n)$ gives the value of the $n$-th cyclotomic polynomial evaluated at $n$: $$a(n) = |\Phi_n(n)|$$

References:

A070518

namespace OeisA70518open Polynomial

Value of the $n$-th cyclotomic polynomial at $n$.

noncomputable def a (n : ℕ) : ℕ :=
  ((cyclotomic n ℤ).eval (n : ℤ)).natAbs

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 0 := by⊢ a 1 = 0
  dsimp [a]⊢ (eval 1 (cyclotomic 1 ℤ)).natAbs = 0
  rw [cyclotomic_one⊢ (eval 1 (X - 1)).natAbs = 0⊢ (eval 1 (X - 1)).natAbs = 0]⊢ (eval 1 (X - 1)).natAbs = 0
  simpAll goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 3 := by⊢ a 2 = 3
  dsimp [a]⊢ (eval 2 (cyclotomic 2 ℤ)).natAbs = 3
  rw [cyclotomic_two⊢ (eval 2 (X + 1)).natAbs = 3⊢ (eval 2 (X + 1)).natAbs = 3]⊢ (eval 2 (X + 1)).natAbs = 3
  simpAll goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 13 := by⊢ a 3 = 13
  dsimp [a]⊢ (eval 3 (cyclotomic 3 ℤ)).natAbs = 13
  have : Fact (Nat.Prime 3) := ⟨by⊢ Nat.Prime 3this:Fact (Nat.Prime 3)⊢ (eval 3 (cyclotomic 3 ℤ)).natAbs = 13 decideAll goals completed! 🐙this:Fact (Nat.Prime 3)⊢ (eval 3 (cyclotomic 3 ℤ)).natAbs = 13⟩this:Fact (Nat.Prime 3)⊢ (eval 3 (cyclotomic 3 ℤ)).natAbs = 13
  rw [cyclotomic_prime ℤ 3this:Fact (Nat.Prime 3)⊢ (eval 3 (∑ i ∈ Finset.range 3, X ^ i)).natAbs = 13this:Fact (Nat.Prime 3)⊢ (eval 3 (∑ i ∈ Finset.range 3, X ^ i)).natAbs = 13]this:Fact (Nat.Prime 3)⊢ (eval 3 (∑ i ∈ Finset.range 3, X ^ i)).natAbs = 13
  simp [Finset.sum_range_succ]All goals completed! 🐙

Value of the sequence a at 5.


@[category test, AMS 11]
theorem a_5 : a 5 = 781 := by⊢ a 5 = 781
  dsimp [a]⊢ (eval 5 (cyclotomic 5 ℤ)).natAbs = 781
  have : Fact (Nat.Prime 5) := ⟨by⊢ Nat.Prime 5this:Fact (Nat.Prime 5)⊢ (eval 5 (cyclotomic 5 ℤ)).natAbs = 781 decideAll goals completed! 🐙this:Fact (Nat.Prime 5)⊢ (eval 5 (cyclotomic 5 ℤ)).natAbs = 781⟩this:Fact (Nat.Prime 5)⊢ (eval 5 (cyclotomic 5 ℤ)).natAbs = 781
  rw [cyclotomic_prime ℤ 5this:Fact (Nat.Prime 5)⊢ (eval 5 (∑ i ∈ Finset.range 5, X ^ i)).natAbs = 781this:Fact (Nat.Prime 5)⊢ (eval 5 (∑ i ∈ Finset.range 5, X ^ i)).natAbs = 781]this:Fact (Nat.Prime 5)⊢ (eval 5 (∑ i ∈ Finset.range 5, X ^ i)).natAbs = 781
  simp [Finset.sum_range_succ]All goals completed! 🐙

open scoped Classical in

$a(28341)$ is divisible by $283411^2$. What is the next $n$ such that $a(n)$ is not squarefree?

@[category research open, AMS 11]
theorem conjecture :
    answer(sorry) =
      if h : ∃ n, 28341 < n ∧ ¬ Squarefree (a n) then
        some (sInf {n | 28341 < n ∧ ¬ Squarefree (a n)})
      else
        none := by⊢ sorry = if h : ∃ n, 28341 < n ∧ ¬Squarefree (a n) then some (sInf {n | 28341 < n ∧ ¬Squarefree (a n)}) else none
  sorryAll goals completed! 🐙

end OeisA70518