80170.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Conjecture relating two characterizations of a set of integers.

Informal Statement: For an integer $k \ge 2$, the following are equivalent:

The greatest common divisor of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.
Writing prime factorization of $k$ as $k = \prod p_i^{e_i}$, and let $P = \max_i p_i^{e_i}$, one has $k / P > P$.

This conjecture asserts that the sequence defined by 1. is obtained by taking 1 off each number in the sequence defined by 2.

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def OeisA80170.A The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$ line 50 source
Source documentation
The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def A (k : ℕ) : Prop :=
…
```
def OeisA80170.B Let $P$ be the largest prime power dividing k line 57 source
Source documentation
Let $P$ be the largest prime power dividing k. Then $k / P > P$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Option.getD, Finset.max, Finset.filter, IsPrimePow, Nat.divisors, OfNat.ofNat, GT.gt, HDiv.hDiv
7 instances, coercions and auxiliary constants left out.
```
def B (k : ℕ) : Prop :=
…
```
theorem OeisA80170.a_2 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, OeisA80170.A, OfNat.ofNat, Nat
Proof uses: id, of_decide_eq_true, Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA80170.a_3 line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, OeisA80170.A, OfNat.ofNat, Nat
Proof uses: id, of_decide_eq_true, Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA80170.a_4 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, OeisA80170.A, OfNat.ofNat, Nat
Proof uses: id, of_decide_eq_true, Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA80170.a_5 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, OeisA80170.A, OfNat.ofNat, Nat
Proof uses: id, of_decide_eq_true, Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA80170.a_6 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, OeisA80170.A, OfNat.ofNat, Nat
Proof uses: id, of_decide_eq_true, Eq, Finset.gcd, Finset.range, Nat.choose, HMul.hMul, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA80170.gcdCondition_iff_primePowerCondition sorry line 90 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Iff, OeisA80170.A, OeisA80170.B, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

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