108866.lean`

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L4DEL: Lean4 Domain Explanation Language

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

8 declarations, 1 contain sorry, 0 declared as axioms

Numerator of $\sum_{k=1}^n 2^k/k$.

Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime.

References:

A108866

Imports (1)

FormalConjecturesUtil

def OeisA108866.a The primary defining sequence a line 39 source
Source documentation
The primary defining sequence a. a n is the numerator of $\sum_{k=1}^n \frac{2^k}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.natAbs, Rat.num, Finset.sum, Rat, Finset.range, HDiv.hDiv, HPow.hPow, OfNat.ofNat, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA108866.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 46 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108866.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by decide
…
```
Mentions: OeisA108866.a
theorem OeisA108866.a_1 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108866.a, OfNat.ofNat
Proof uses: id, Mathlib.Meta.NormNum.isNat_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.range, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Mathlib.Meta.NormNum.isNat_natAbs_pos, Mathlib.Meta.NormNum.IsInt.to_isNat, Int, Mathlib.Meta.NormNum.isInt_ratNum, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, inferInstance, DivisionRing, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Eq.rec and 28 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA108866.a_2 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108866.a, OfNat.ofNat
Proof uses: id, Mathlib.Meta.NormNum.isNat_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.range, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Mathlib.Meta.NormNum.isNat_natAbs_pos, Mathlib.Meta.NormNum.IsInt.to_isNat, Int, Mathlib.Meta.NormNum.isInt_ratNum, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, inferInstance, DivisionRing, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Eq.rec and 32 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA108866.a_3 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108866.a, OfNat.ofNat
Proof uses: id, Mathlib.Meta.NormNum.isNat_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.range, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Mathlib.Meta.NormNum.isNat_natAbs_pos, Mathlib.Meta.NormNum.IsInt.to_isNat, Int, Mathlib.Meta.NormNum.isInt_ratNum, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, inferInstance, DivisionRing, Eq.rec, Finset.cons and 35 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA108866.a_4 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108866.a, OfNat.ofNat
Proof uses: id, Mathlib.Meta.NormNum.isNat_eq_true, Int.natAbs, Rat.num, Finset.sum, Rat, Finset.range, HDiv.hDiv, HPow.hPow, HAdd.hAdd, Mathlib.Meta.NormNum.isNat_natAbs_pos, Mathlib.Meta.NormNum.IsInt.to_isNat, Int, Mathlib.Meta.NormNum.isInt_ratNum, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, inferInstance, DivisionRing, Eq.rec, Finset.cons and 37 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA108866.ratExpression The rational number inside the numerator function in the conjecture line 64 source
Source documentation
The rational number inside the numerator function in the conjecture. $$ -\frac{2}{n} + \sum_{k=1}^n \frac{2^k}{k} $$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: ite, GT.gt, OfNat.ofNat, HAdd.hAdd, HDiv.hDiv, Neg.neg, Finset.sum, Finset.range, HPow.hPow
15 instances, coercions and auxiliary constants left out.
```
noncomputable def ratExpression (n : ℕ) : ℚ :=
…
```
theorem OeisA108866.conjecture Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime sorry line 76 source
Source documentation
Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Iff, Int.ModEq, HPow.hPow, Int, Rat.num, OeisA108866.ratExpression, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture {n : ℕ} (hn : n > 3) :
    (ratExpression n).num ≡ 0 [ZMOD (n^2 : ℤ)] ↔ n.Prime := by
…
```
Mentions: OeisA108866.ratExpression