160324.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

13 declarations, 5 contain sorry, 0 declared as axioms

Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers

$$a(n) = |\{(x, y, z) \in \mathbb{N}^3 : x^2 + p_5(y) + p_6(z) = n\}|$$

References:

A160324

Imports (1)

FormalConjecturesUtil

def OeisA160324.polygonalNumber $p_k(x) = \frac{(k-2)x(x-1)}{2} + x$ is the $x$-th $k$-gonal number line 33 source
Source documentation
$p_k(x) = \frac{(k-2)x(x-1)}{2} + x$ is the $x$-th $k$-gonal number.

no axioms what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, HMul.hMul, HSub.hSub, OfNat.ofNat, HDiv.hDiv
9 instances, coercions and auxiliary constants left out.
```
def polygonalNumber (k x : ℕ) : ℕ :=
…
```
def OeisA160324.pentagonal $p_5(y) = \frac{3y^2 - y}{2}$ is the $y$-th pentagonal number line 37 source
Source documentation
$p_5(y) = \frac{3y^2 - y}{2}$ is the $y$-th pentagonal number.

no axioms what this means
Statement uses: Nat
Proof uses: OeisA160324.polygonalNumber, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def pentagonal (y : ℕ) : ℕ := polygonalNumber 5 y
```
Mentions: OeisA160324.polygonalNumber
def OeisA160324.hexagonal $p_6(z) = 2z^2 - z$ is the $z$-th hexagonal number line 40 source
Source documentation
$p_6(z) = 2z^2 - z$ is the $z$-th hexagonal number.

no axioms what this means
Statement uses: Nat
Proof uses: OeisA160324.polygonalNumber, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def hexagonal (z : ℕ) : ℕ := polygonalNumber 6 z
```
Mentions: OeisA160324.polygonalNumber
def OeisA160324.a Number of representations of $n$ as sum of square, pentagonal, and hexagonal numbers line 43 source
Source documentation
Number of representations of $n$ as sum of square, pentagonal, and hexagonal numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, Nat.sqrt, OfNat.ofNat, Finset.sum, Finset.range, ite, Eq, HPow.hPow, OeisA160324.pentagonal, OeisA160324.hexagonal
9 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA160324.a_0 Value of the sequence a at 0 line 52 source
Source documentation
Value of the sequence a at 0.

extra axioms: _private.0.OeisA160324.a_0._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA160324.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 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 decide +native
…
```
Mentions: OeisA160324.a
theorem OeisA160324.a_1 Value of the sequence a at 1 line 56 source
Source documentation
Value of the sequence a at 1.

extra axioms: _private.0.OeisA160324.a_1._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA160324.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 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 = 3 := by decide +native
…
```
Mentions: OeisA160324.a
theorem OeisA160324.a_2 Value of the sequence a at 2 line 60 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA160324.a_2._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA160324.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 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 = 3 := by decide +native
…
```
Mentions: OeisA160324.a
theorem OeisA160324.a_3 Value of the sequence a at 3 line 64 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA160324.a_3._native.decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA160324.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 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 = 1 := by decide +native
…
```
Mentions: OeisA160324.a
theorem OeisA160324.conjecture1 In April 2009, Zhi-Wei Sun conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \dots$ sorry line 70 source
Source documentation
In April 2009, Zhi-Wei Sun conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \dots$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA160324.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 (n : ℕ) : 0 < a n := by
…
```
Mentions: OeisA160324.a
theorem OeisA160324.conjecture2 For each integer sorry line 80 source
Source documentation
For each integer $m > 2$, any natural number $n$ can be written in the form $p_{m+1}(x_1) + \cdots + p_{2m}(x_m)$ with $x_1, \dots, x_m$ nonnegative integers, where $p_k(x) = (k-2)x(x-1)/2 + x$ ($x=0,1,2,\dots$) are $k$-gonal numbers.
- Zhi-Wei Sun, Aug 15 2009
depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Exists, Fin, Eq, Finset.sum, Finset.univ, OeisA160324.polygonalNumber, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (m : ℕ) (hm : m > 2) (n : ℕ) :
    ∃ x : Fin m → ℕ, n = ∑ i : Fin m, polygonalNumber (m + (i : ℕ) + 1) (x i) := by
…
```
Mentions: OeisA160324.polygonalNumber
theorem OeisA160324.conjecture3 The sequence contains every positive integer sorry line 89 source
Source documentation
The sequence contains every positive integer.
- Zhi-Wei Sun, Sep 04 2009
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, Eq, OeisA160324.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 (k : ℕ) (hk : 0 < k) : ∃ n : ℕ, a n = k := by
…
```
Mentions: OeisA160324.a
theorem OeisA160324.conjecture4 Conjecture (Zhi-Wei Sun, Aug 21 2009): For any integer sorry line 99 source
Source documentation
Conjecture (Zhi-Wei Sun, Aug 21 2009): For any integer $m > 2$, each natural number $n$ can be expressed as $p_{m+1}(x_1) + p_{m+2}(x_2) + p_{m+3}(x_3) + r$ with $x_1, x_2, x_3 \in \mathbb{N}$ and $r \in \{0, \dots, m-3\}$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, And, LE.le, HSub.hSub, Eq, HAdd.hAdd, OeisA160324.polygonalNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture4 (m : ℕ) (hm : 2 < m) (n : ℕ) :
    ∃ x1 x2 x3 r : ℕ, r ≤ m - 3 ∧
      n = polygonalNumber (m + 1) x1 + polygonalNumber (m + 2) x2 + polygonalNumber (m + 3) x3 + r := by
…
```
Mentions: OeisA160324.polygonalNumber
theorem OeisA160324.conjecture5 Conjecture (Zhi-Wei Sun, Aug 21 2009): For each integer sorry line 110 source
Source documentation
Conjecture (Zhi-Wei Sun, Aug 21 2009): For each integer $m > 2$, all sufficiently large integers $n$ can be expressed in the form $p_{m+1}(x_1) + p_{m+2}(x_2) + p_{m+3}(x_3)$ with $x_1, x_2, x_3 \in \mathbb{N}$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Filter.Eventually, Exists, Eq, HAdd.hAdd, OeisA160324.polygonalNumber, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture5 (m : ℕ) (hm : 2 < m) :
    ∀ᶠ n in Filter.atTop,
      ∃ x1 x2 x3 : ℕ, n = polygonalNumber (m + 1) x1 + polygonalNumber (m + 2) x2 + polygonalNumber (m + 3) x3 := by
…
```
Mentions: OeisA160324.polygonalNumber