Sum of three cubes

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-/

import FormalConjecturesUtil

/-!
# 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](https://en.wikipedia.org/wiki/Sums_of_three_cubes)
 - [mathoverflow/100324](https://mathoverflow.net/a/100324)
asked by user [*David Feldman*](https://mathoverflow.net/users/10909/david-feldman)
-/

namespace SumOfThreeCubes

variable {R : Type*} [Ring R]

/-- The predicate that `n : R` is a sum of three cubes. -/
def IsSumOfThreeCubes (n : R) : Prop :=
  ∃ x y z : R, n = x^3 + y^3 + z^3

@[category test, AMS 11]
theorem isSumOfThreeCubes_2 : IsSumOfThreeCubes (2 : ℤ) :=
  ⟨1, 1, 0, by norm_num⟩

@[category test, AMS 11]
theorem isSumOfThreeCubes_33 : IsSumOfThreeCubes (33 : ℤ) :=
  ⟨8866128975287528, -8778405442862239, -2736111468807040, by norm_num⟩

@[category test, AMS 11]
theorem isSumOfThreeCubes_42 : IsSumOfThreeCubes (42 : ℤ) :=
  ⟨-80538738812075974, 80435758145817515, 12602123297335631, by norm_num⟩

@[category test, AMS 11]
theorem mod_9_of_isSumOfThreeCubes (n : ℤ) (hn : IsSumOfThreeCubes n) :
    ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9]) := by
  rw [show (9 : ℤ) = (9 : ℕ) from rfl, ← ZMod.intCast_eq_intCast_iff _ _ 9,
    ← ZMod.intCast_eq_intCast_iff _ _ 9]
  obtain ⟨x, y, z, hn⟩ := hn
  replace hn := congrArg (fun x : ℤ ↦ (x : ZMod 9)) hn
  simp only [Int.cast_add, Int.cast_pow] at hn
  generalize (n : ZMod 9) = n at *
  generalize (x : ZMod 9) = x at *
  generalize (y : ZMod 9) = y at *
  generalize (z : ZMod 9) = z at *
  decide +revert

/--
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
-/
@[category research solved, AMS 11]
theorem isSumOfThreeCubesRat_any (r : ℚ) : IsSumOfThreeCubes r := by
  by_cases h : r = 0
  · exact ⟨0, 0, 0, by norm_num; exact h⟩
  · let x := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)
    let y := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2
    let z := (r ^ 2 + 6 * r + 3) * (- r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))
    use x, y, z
    simp only [x, y, z]
    field_simp
    ring


/-- 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`. -/
@[category research open, AMS 11]
theorem isSumOfThreeCubes_iff_mod_9 :
    answer(sorry) ↔ ∀ n : ℤ, IsSumOfThreeCubes n ↔ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9]) := by
  sorry

end SumOfThreeCubes

• Formalization source ↗

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