281976.lean`

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11 declarations, 1 contain sorry, 0 declared as axioms

Sum of four squares with square conditions

Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares.

Zhi-Wei Sun has offered a $2,400 prize for the first proof.

References:

A281976
Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190. https://doi.org/10.1016/j.jnt.2016.11.008
Z.-W. Sun, "Restricted sums of four squares," arXiv:1701.05868 [math.NT], 2017. https://arxiv.org/abs/1701.05868

Imports (1)

FormalConjecturesUtil

def OeisA281976.A The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers, $z \leq w$, such that both $x$ and $x + 24y$ are squares line 42 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, $z \leq w$, such that both $x$ and $x + 24y$ are squares.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, LE.le, IsSquare, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA281976.a_0 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, IsSquare, HMul.hMul, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow and 8 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_1 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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_le_true, Bool, Bool.true and 1 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_2 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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_le_true, Bool, Bool.true and 1 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_3 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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_le_true, Bool, Bool.true and 1 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_4 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.isNat_le_true, Bool and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_8 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.isNat_le_true, Bool and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_12 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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_le_true, Bool and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_23 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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_le_true, Bool and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.a_24 line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA281976.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, Eq, HAdd.hAdd, HPow.hPow, LE.le, 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.isNat_le_true, Bool and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA281976.conjecture Zhi-Wei Sun's Conjecture (A281976): Any integer sorry line 86 source
Source documentation
Zhi-Wei Sun's Conjecture (A281976): Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares.

depends on sorry what this means
Statement uses: Nat, OeisA281976.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: OeisA281976.A