309132.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Characterization of Carmichael numbers via squarefree denominators $a(n)$

$a(n)$ is the denominator of $F(n) = \operatorname{num}(B_{n-1})/n + \operatorname{den}(B_{n-1})/n^2$.

A composite number $n$ has squarefree $a(n)$ if and only if $n$ is a Carmichael number.

References:

A309132
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
A027641
A027642

Imports (1)

FormalConjecturesUtil

def OeisA309132.a $a(n)$ is the denominator of $F(n)$ = A027641(n-1)/n + A027642(n-1)/n^2 line 44 source
Source documentation
$a(n)$ is the denominator of $F(n)$ = A027641(n-1)/n + A027642(n-1)/n^2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul
14 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA309132.IsCarmichaelNumber Definition of a Carmichael number $n$: a composite number s.t line 54 source
Source documentation
Definition of a Carmichael number $n$: a composite number s.t. $b^{n-1} \equiv 1 \pmod n$ for all $b$ coprime to $n$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Not, Nat.Prime, GT.gt, OfNat.ofNat, Eq, Nat.gcd, Nat.ModEq, HPow.hPow, HSub.hSub
8 instances, coercions and auxiliary constants left out.
```
def IsCarmichaelNumber (n : ℕ) : Prop :=
…
```
def OeisA309132.IsComposite Helper definition for "composite number" line 58 source
Source documentation
Helper definition for "composite number"

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Not, Nat.Prime, GT.gt, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.
```
def IsComposite (n : ℕ) : Prop := ¬ Nat.Prime n ∧ n > 1
…
```
lemma OeisA309132.a_1 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA309132.a, OfNat.ofNat
Proof uses: id, of_eq_true, ite, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite.congr_simp, False, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat and 31 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA309132.a_2 line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA309132.a, OfNat.ofNat
Proof uses: id, of_eq_true, ite, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite.congr_simp, False, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat and 49 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA309132.a_3 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA309132.a, OfNat.ofNat
Proof uses: id, of_eq_true, ite, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite.congr_simp, False, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat and 45 more
59 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA309132.a_4 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA309132.a, OfNat.ofNat
Proof uses: id, of_eq_true, ite, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite_congr, False, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat and 44 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA309132.a_5 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA309132.a, OfNat.ofNat
Proof uses: id, of_eq_true, ite, Rat, bernoulli, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Rat.num, Rat.den, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite.congr_simp, False, eq_false, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat and 51 more
67 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA309132.carmichael_iff_squarefree_a sorry line 85 source
depends on sorry what this means
Statement uses: Nat, Iff, And, OeisA309132.IsComposite, Squarefree, OeisA309132.a, OeisA309132.IsCarmichaelNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.