46969.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

12 declarations, 2 contain sorry, 0 declared as axioms

Denominators of coefficients in Stirling's expansion for $\log(\Gamma(z))$

The $n$-th term is the denominator of $\frac{B_{2n}}{2n(2n-1)}$ where $B_{2n}$ is the $2n$-th Bernoulli number.

References:

A046969

Imports (1)

FormalConjecturesUtil

def OeisA46969.a Denominators of coefficients in Stirling's expansion for $\log(\Gamma(z))$ line 34 source
Source documentation
Denominators of coefficients in Stirling's expansion for $\log(\Gamma(z))$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HMul.hMul, HSub.hSub, Rat.den, HDiv.hDiv, Rat, bernoulli
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA46969.a_0 line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.a_1 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Rat.den, HDiv.hDiv, Rat, bernoulli, Inv.inv, congrArg, bernoulli_two, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, inferInstance, DivisionRing, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, HMul.hMul, Eq.refl, Mathlib.Meta.NormNum.isNNRat_inv_pos and 5 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.a_2 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Rat.den, HDiv.hDiv, Rat, bernoulli, bernoulli', congrArg, bernoulli_eq_bernoulli'_of_ne_one, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, Neg.neg, bernoulli'_four, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ratDen, Int.negOfNat and 16 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA46969.bernoulli'_six line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, bernoulli', OfNat.ofNat, Nat, HDiv.hDiv
Proof uses: Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, bernoulli'_eq_zero_of_odd, Odd, LT.lt, Eq.mpr, HSub.hSub, Finset.sum, Finset.range, HMul.hMul, HAdd.hAdd, congrArg, bernoulli'_def, of_eq_true, Eq.trans and 64 more
93 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.a_3 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Rat.den, HDiv.hDiv, Rat, bernoulli, bernoulli', congrArg, bernoulli_eq_bernoulli'_of_ne_one, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, OeisA46969.bernoulli'_six, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat and 12 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA46969.bernoulli'_eight line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, bernoulli', OfNat.ofNat, Nat, HDiv.hDiv, Neg.neg
Proof uses: Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, bernoulli'_three, bernoulli'_eq_zero_of_odd, Odd, LT.lt, Eq.mpr, HSub.hSub, Finset.sum, Finset.range, HMul.hMul, HAdd.hAdd, congrArg, bernoulli'_def, of_eq_true and 64 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.a_4 line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Rat.den, HDiv.hDiv, Rat, bernoulli, bernoulli', congrArg, bernoulli_eq_bernoulli'_of_ne_one, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, Neg.neg, OeisA46969.bernoulli'_eight, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ratDen, Int.negOfNat and 16 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA46969.bernoulli'_ten line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, bernoulli', OfNat.ofNat, Nat, HDiv.hDiv
Proof uses: Nat.choose, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, bernoulli'_three, bernoulli'_eq_zero_of_odd, Odd, LT.lt, Eq.mpr, HSub.hSub, Finset.sum, Finset.range, HMul.hMul, HAdd.hAdd, congrArg, bernoulli'_def, of_eq_true and 67 more
93 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.a_5 line 112 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA46969.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Rat.den, HDiv.hDiv, Rat, bernoulli, bernoulli', congrArg, bernoulli_eq_bernoulli'_of_ne_one, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, OeisA46969.bernoulli'_ten, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat and 12 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA46969.conjecture1 Conjecture I: if $n > 2$, then $\frac{a(\text{A005382}(n))}{12}$ is prime, where A005382 is the sequence of primes $p$ such that $2p-1$ is also prime sorry line 125 source
Source documentation
Conjecture I: if $n > 2$, then $\frac{a(\text{A005382}(n))}{12}$ is prime, where A005382 is the sequence of primes $p$ such that $2p-1$ is also prime. Since $\text{A005382}(1) = 2$, $\text{A005382}(2) = 3$ and $\text{A005382}(3) = 7$, this says that $\frac{a(p)}{12}$ is prime for every prime $p > 3$ such that $2p-1$ is also prime.
- Lorenzo Sauras Altuzarra, Oct 13 2020
depends on sorry what this means
Statement uses: Nat, Nat.Prime, HSub.hSub, HMul.hMul, OfNat.ofNat, LT.lt, HDiv.hDiv, OeisA46969.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (p : ℕ) (hp : p.Prime) (hp' : (2 * p - 1).Prime) (h3 : 3 < p) :
    (a p / 12).Prime := by
…
```
Mentions: OeisA46969.a
theorem OeisA46969.conjecture2 sorry line 138 source
depends on sorry what this means
Statement uses: Not, Nat, LE.le, OfNat.ofNat, Dvd.dvd, OeisA46969.a, Nat.Prime, HDiv.hDiv, HSub.hSub, HAdd.hAdd, And, Int
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

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