287616.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Sum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number

Any nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$ nonnegative integers.

Zhi-Wei Sun offered a USD 135 prize for the first proof of this conjecture. Cao, Guo, Qiu, Feng and Gao proved it in [CGQFG26].

References:

A287616
Zhi-Wei Sun, "Universal sums of three quadratic polynomials", arXiv:1502.03056 [math.NT]
[CGQFG26] Y. Cao, D. Guo, R. Qiu, R. Feng and X.-S. Gao, "Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number", arXiv:2606.26035

Imports (1)

FormalConjecturesUtil

def OeisA287616.A The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for nonnegative integers $x, y, z$ line 43 source
Source documentation
The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for nonnegative integers $x, y, z$.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA287616.a_0 line 47 source
standard axioms: propext what this means
Statement uses: OeisA287616.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Meta.NormNum.isNat_mul
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA287616.a_1 line 51 source
standard axioms: propext what this means
Statement uses: OeisA287616.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Meta.NormNum.isNat_mul
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA287616.a_2 line 55 source
standard axioms: propext what this means
Statement uses: OeisA287616.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Meta.NormNum.isNat_mul
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA287616.a_3 line 59 source
standard axioms: propext what this means
Statement uses: OeisA287616.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Meta.NormNum.isNat_mul
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA287616.a_4 line 63 source
standard axioms: propext what this means
Statement uses: OeisA287616.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HDiv.hDiv, HMul.hMul, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_natDiv, Mathlib.Meta.NormNum.isNat_mul
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA287616.conjecture Zhi-Wei Sun's Conjecture (A287616): Any nonnegative integer can be written as the sum of a triangular number sorry line 74 source
Source documentation
Zhi-Wei Sun's Conjecture (A287616): Any nonnegative integer can be written as the sum of a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized heptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.
This was proved in [CGQFG26].

depends on sorry what this means
Statement uses: Nat, OeisA287616.A
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 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) : A n := by
…
```
Mentions: OeisA287616.A