3161.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

A binomial coefficient sum

A binomial coefficient sum: $$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \left( \binom{n}{k} - \binom{n}{k-1} \right)^3$$ where $\binom{n}{-1} = 0$.

References:

A003161

Imports (1)

FormalConjecturesUtil

def OeisA3161.a A binomial coefficient sum: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} (\binom{n}{k} - \binom{n}{k-1})^3$ line 36 source
Source documentation
A binomial coefficient sum: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} (\binom{n}{k} - \binom{n}{k-1})^3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, Int, HSub.hSub, Nat.choose, ite, Eq, Int.toNat, HPow.hPow
17 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA3161.b Auxiliary sequence $b(n) = a(2n-1)$ line 42 source
Source documentation
Auxiliary sequence $b(n) = a(2n-1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA3161.a, HSub.hSub, HMul.hMul, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℕ :=
…
```
theorem OeisA3161.a_0 Value of the sequence a at 0 line 47 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA3161.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 a_0 : a 0 = 1 := by rfl
…
```
Mentions: OeisA3161.a
theorem OeisA3161.a_1 Value of the sequence a at 1 line 51 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA3161.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 a_1 : a 1 = 1 := by rfl
…
```
Mentions: OeisA3161.a
theorem OeisA3161.a_2 Value of the sequence a at 2 line 55 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA3161.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 a_2 : a 2 = 2 := by rfl
…
```
Mentions: OeisA3161.a
theorem OeisA3161.a_3 Value of the sequence a at 3 line 59 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA3161.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 a_3 : a 3 = 9 := by rfl
…
```
Mentions: OeisA3161.a
theorem OeisA3161.a_4 Value of the sequence a at 4 line 63 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA3161.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 a_4 : a 4 = 36 := by rfl
…
```
Mentions: OeisA3161.a
theorem OeisA3161.conjecture Let $b(n) = a(2n-1)$ sorry line 70 source
Source documentation
Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \equiv b(n p^{k-1}) \pmod{p^{3k}}$ holds for positive integers $n$ and $k$ and all primes $p \ge 5$.
- Zhi-Wei Sun, Nov 16 2019
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, LE.le, Int.ModEq, HPow.hPow, Int, HMul.hMul, OeisA3161.b, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n k p : ℕ) (hn : 0 < n) (hk : 0 < k) (hp : p.Prime) (hp_ge : 5 ≤ p) :
    (b (n * p ^ k) : ℤ) ≡ (b (n * p ^ (k - 1)) : ℤ) [ZMOD (p : ℤ) ^ (3 * k)] := by
…
```
Mentions: OeisA3161.b