280831.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

The 1680-Conjecture

Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

Zhi-Wei Sun has offered a prize of 1,680 RMB for the first proof.

References:

A280831
Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190.
Z.-W. Sun, "Refining Lagrange's four-square theorem," arXiv:1604.06723 [math.NT], 2016.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def OeisA280831.A The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square line 40 source
Source documentation
The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, IsSquare, HMul.hMul
9 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA280831.a_0 line 44 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_1 line 48 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_2 line 52 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_3 line 56 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_4 line 60 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.IsNatPowT.trans, Nat.pow, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_7 line 64 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.isNat_mul
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.a_95 line 68 source
standard axioms: propext what this means
Statement uses: OeisA280831.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, IsSquare, HMul.hMul, And.intro, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.IsNatPowT.trans, Nat.pow, Mathlib.Meta.NormNum.isNat_mul and 1 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA280831.conjecture Documented declaration: conjecture sorry line 76 source
Source documentation
Zhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

depends on sorry what this means
Statement uses: Nat, OeisA280831.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: OeisA280831.A