70518.lean`

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

6 declarations, 1 contain sorry, 0 declared as axioms

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

Imports (1)

FormalConjecturesUtil

def OeisA70518.a Value of the $n$-th cyclotomic polynomial at $n$ line 37 source
Source documentation
Value of the $n$-th cyclotomic polynomial at $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic
4 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA70518.a_1 Value of the sequence a at 1 line 42 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA70518.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, HSub.hSub, Polynomial, Polynomial.X, congrArg, Polynomial.cyclotomic_one, of_eq_true, Eq.trans, True, congrFun', Polynomial.eval_sub, congr, Polynomial.eval_X, Polynomial.eval_one, sub_self and 1 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by
…
```
Mentions: OeisA70518.a
theorem OeisA70518.a_2 Value of the sequence a at 2 line 49 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA70518.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, HAdd.hAdd, Polynomial, Polynomial.X, congrArg, Polynomial.cyclotomic_two, of_eq_true, Eq.trans, True, congrFun', Polynomial.eval_add, congr, Polynomial.eval_X, Polynomial.eval_one, eq_self
19 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 3 := by
…
```
Mentions: OeisA70518.a
theorem OeisA70518.a_3 Value of the sequence a at 3 line 56 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA70518.a, OfNat.ofNat
Proof uses: id, Fact, Nat.Prime, Fact.mk, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, Finset.sum, Polynomial, Finset.range, HPow.hPow, Polynomial.X, congrArg and 18 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 13 := by
…
```
Mentions: OeisA70518.a
theorem OeisA70518.a_5 Value of the sequence a at 5 line 64 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA70518.a, OfNat.ofNat
Proof uses: id, Fact, Nat.Prime, Fact.mk, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Int.natAbs, Polynomial.eval, Int, Polynomial.cyclotomic, Finset.sum, Polynomial, Finset.range, HPow.hPow, Polynomial.X, congrArg and 18 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 781 := by
…
```
Mentions: OeisA70518.a
theorem OeisA70518.conjecture $a(28341)$ is divisible by $283411^2$ sorry line 75 source
Source documentation
$a(28341)$ is divisible by $283411^2$. What is the next $n$ such that $a(n)$ is not squarefree?

depends on sorry what this means
Statement uses: Eq, Option, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, dite, Exists, And, LT.lt, Not, Squarefree, OeisA70518.a, Classical.propDecidable, Option.some, InfSet.sInf and 2 more
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    answer(sorry) =
      if h : ∃ n, 28341 < n ∧ ¬ Squarefree (a n) then
        some (sInf {n | 28341 < n ∧ ¬ Squarefree (a n)})
      else
        none := by
…
```
Mentions: OeisA70518.a