3162.lean`

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L4DEL: Lean4 Domain Explanation Language

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

9 declarations, 2 contain sorry, 0 declared as axioms

A binomial coefficient summation

A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$, where for a positive integer $r$ we define $$S(r,n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \left( \binom{n}{k} - \binom{n}{k-1} \right)^r$$ with $\binom{n}{-1} = 0$.

References:

A003162
H. W. Gould, Problem E2384 (proposal), Amer. Math. Monthly, 81 (1974), 170-171

Imports (1)

FormalConjecturesUtil

def OeisA3162.a A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$ line 38 source
Source documentation
A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, HSub.hSub, Nat.choose, ite, Eq, HPow.hPow
18 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℚ :=
…
```
def OeisA3162.b Auxiliary sequence $b(n) = a(2n-1)$ line 46 source
Source documentation
Auxiliary sequence $b(n) = a(2n-1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: OeisA3162.a, HSub.hSub, HMul.hMul, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℚ :=
…
```
theorem OeisA3162.a_0 Value of the sequence a at 0 line 55 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA3162.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HDiv.hDiv, Finset.sum, Singleton.singleton, Finset, HPow.hPow, HSub.hSub, Nat.choose, ite, Eq.trans, True, congrFun', congrArg, congr, Finset.sum_singleton, Nat.choose_self, Nat.cast_one, ite_congr, eq_true and 15 more
49 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 eval_a
…
```
Mentions: OeisA3162.a
theorem OeisA3162.a_1 Value of the sequence a at 1 line 59 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA3162.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HDiv.hDiv, Finset.sum, Singleton.singleton, Finset, HPow.hPow, HSub.hSub, Nat.choose, ite, Eq.trans, True, congrFun', congrArg, congr, Finset.sum_singleton, HAdd.hAdd, Nat.choose_succ_self_right, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add and 17 more
54 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 eval_a
…
```
Mentions: OeisA3162.a
theorem OeisA3162.a_2 Value of the sequence a at 2 line 63 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA3162.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HDiv.hDiv, Finset.sum, Finset.range, HPow.hPow, HSub.hSub, Nat.choose, ite, Eq.trans, True, congrFun', congrArg, congr, HAdd.hAdd, Finset.sum_range_succ, Singleton.singleton, Finset, Finset.sum_singleton, Nat.choose_zero_right and 29 more
67 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 = 1 := by eval_a
…
```
Mentions: OeisA3162.a
theorem OeisA3162.a_3 Value of the sequence a at 3 line 67 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA3162.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HDiv.hDiv, Finset.sum, Finset.range, HPow.hPow, HSub.hSub, Nat.choose, ite, Eq.trans, True, congrFun', congrArg, congr, HAdd.hAdd, Finset.sum_range_succ, Singleton.singleton, Finset, Finset.sum_singleton, Nat.choose_zero_right and 40 more
66 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 = 3 := by eval_a
…
```
Mentions: OeisA3162.a
theorem OeisA3162.a_4 Value of the sequence a at 4 line 71 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA3162.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HDiv.hDiv, Finset.sum, Finset.range, HPow.hPow, HSub.hSub, Nat.choose, ite, Eq.trans, True, congrFun', congrArg, congr, HAdd.hAdd, Finset.sum_range_succ, Singleton.singleton, Finset, Finset.sum_singleton, Nat.choose_zero_right and 40 more
66 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 = 6 := by eval_a
…
```
Mentions: OeisA3162.a
theorem OeisA3162.a_is_integer $a(n)$ is an integer for all $n \ge 0$ sorry line 80 source
Source documentation
$a(n)$ is an integer for all $n \ge 0$.
Gould proposed this as Monthly Problem E2384. The 1974 reference in the module docstring is that proposal, not a solution to it.

depends on sorry what this means
Statement uses: Nat, Eq, Rat.den, OeisA3162.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_is_integer (n : ℕ) : (a n).den = 1 := by
…
```
Mentions: OeisA3162.a
theorem OeisA3162.conjecture Let $b(n) = a(2n-1)$ sorry line 89 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, Rat.num, OeisA3162.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)).num ≡ (b (n * p ^ (k - 1))).num [ZMOD (p : ℤ) ^ (3 * k)] := by
…
```
Mentions: OeisA3162.b