SumOfThreeCubes.lean`

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7 declarations, 1 contain sorry, 0 declared as axioms

Sum of three cubes

An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9.

References:

Wikipedia
mathoverflow/100324 asked by user David Feldman

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def SumOfThreeCubes.IsSumOfThreeCubes The predicate that n : R is a sum of three cubes line 39 source
Source documentation
The predicate that n : R is a sum of three cubes.

no axioms what this means
Statement uses: Ring
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, Nat, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def IsSumOfThreeCubes (n : R) : Prop :=
…
```
theorem SumOfThreeCubes.isSumOfThreeCubes_2 line 43 source
standard axioms: propext what this means
Statement uses: SumOfThreeCubes.IsSumOfThreeCubes, Int, OfNat.ofNat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat, 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
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem SumOfThreeCubes.isSumOfThreeCubes_33 line 47 source
standard axioms: Quot.sound, propext what this means
Statement uses: SumOfThreeCubes.IsSumOfThreeCubes, Int, OfNat.ofNat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat, Neg.neg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_add, Int.ofNat, Int.negOfNat, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit1, Mathlib.Meta.NormNum.isInt_pow, Mathlib.Meta.NormNum.isInt_neg and 2 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem SumOfThreeCubes.isSumOfThreeCubes_42 line 51 source
standard axioms: Quot.sound, propext what this means
Statement uses: SumOfThreeCubes.IsSumOfThreeCubes, Int, OfNat.ofNat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Nat, Neg.neg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_add, Int.negOfNat, Int.ofNat, Mathlib.Meta.NormNum.isInt_pow, Mathlib.Meta.NormNum.isInt_neg, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.intPow_negOfNat_bit1, Mathlib.Meta.NormNum.natPow_one, Mathlib.Meta.NormNum.isNat_pow and 2 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem SumOfThreeCubes.mod_9_of_isSumOfThreeCubes line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, SumOfThreeCubes.IsSumOfThreeCubes, Not, Or, Int.ModEq, OfNat.ofNat
Proof uses: Eq.mpr, Nat, id, Eq, congrArg, rfl, ZMod, Eq.symm, propext, ZMod.intCast_eq_intCast_iff, Exists, HAdd.hAdd, HPow.hPow, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mp, Eq.trans and 3 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
theorem SumOfThreeCubes.isSumOfThreeCubesRat_any Any rational number is a sum of three rational cubes line 78 source
Source documentation
Any rational number is a sum of three rational cubes.
First proved by Ryley in 1825, which can be found in [Ri1930]. The below parametrization is brought from the MSE answer [MSE].
[Ri1930] Richmond, H. W. "On Rational Solutions of $x^3 + y^3 + z^3 = R$." Proceedings of the Edinburgh Mathematical Society 2.2 (1930): 92-100. [MSE] Kieren MacMillan, Proving that any rational number can be represented as the sum of the cubes of three rational numbers, https://math.stackexchange.com/q/4480969

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, SumOfThreeCubes.IsSumOfThreeCubes
Proof uses: dite, Eq, OfNat.ofNat, Exists.intro, Exists, HAdd.hAdd, HPow.hPow, Nat, Eq.mpr, id, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.zero_natPow, Not, HDiv.hDiv, HSub.hSub and 124 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem isSumOfThreeCubesRat_any (r : ℚ) : IsSumOfThreeCubes r := by
…
```
Mentions: SumOfThreeCubes.IsSumOfThreeCubes
theorem SumOfThreeCubes.isSumOfThreeCubes_iff_mod_9 An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9 sorry line 93 source
Source documentation
An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9.

depends on sorry what this means
Statement uses: Iff, True, Int, SumOfThreeCubes.IsSumOfThreeCubes, Not, Or, Int.ModEq, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isSumOfThreeCubes_iff_mod_9 :
    answer(sorry) ↔ ∀ n : ℤ, IsSumOfThreeCubes n ↔ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9]) := by
…
```
Mentions: SumOfThreeCubes.IsSumOfThreeCubes