FormalConjectures.Wikipedia.SumOfThreeCubes

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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

mathoverflow/100324 asked by user 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⊢ 2 = 1 ^ 3 + 1 ^ 3 + 0 ^ 3 norm_numAll goals completed! 🐙⟩


@[category test, AMS 11]
theorem isSumOfThreeCubes_33 : IsSumOfThreeCubes (33 : ℤ) :=
  ⟨8866128975287528, -8778405442862239, -2736111468807040, by⊢ 33 = 8866128975287528 ^ 3 + (-8778405442862239) ^ 3 + (-2736111468807040) ^ 3 norm_numAll goals completed! 🐙⟩


@[category test, AMS 11]
theorem isSumOfThreeCubes_42 : IsSumOfThreeCubes (42 : ℤ) :=
  ⟨-80538738812075974, 80435758145817515, 12602123297335631, by⊢ 42 = (-80538738812075974) ^ 3 + 80435758145817515 ^ 3 + 12602123297335631 ^ 3 norm_numAll goals completed! 🐙⟩


@[category test, AMS 11]
theorem mod_9_of_isSumOfThreeCubes (n : ℤ) (hn : IsSumOfThreeCubes n) :
    ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9]) := byn:ℤhn:IsSumOfThreeCubes n⊢ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9])
  rw [show (9 : ℤ) = (9 : ℕ) from rfl, ← ZMod.intCast_eq_intCast_iff _ _ 9,
    ← ZMod.intCast_eq_intCast_iff _ _ 9]n:ℤhn:IsSumOfThreeCubes n⊢ ¬(↑n = ↑4 ∨ ↑n = ↑5)
  obtain ⟨x, y, z, hn⟩ := hnn:ℤx:ℤy:ℤz:ℤhn:n = x ^ 3 + y ^ 3 + z ^ 3⊢ ¬(↑n = ↑4 ∨ ↑n = ↑5)
  replace hn := congrArg (fun x : ℤ ↦ (x : ZMod 9)) hnn:ℤx:ℤy:ℤz:ℤhn:(fun x => ↑x) n = (fun x => ↑x) (x ^ 3 + y ^ 3 + z ^ 3) := congrArg (fun x => ↑x) _fvar.3488⊢ ¬(↑n = ↑4 ∨ ↑n = ↑5)
  simp only [Int.cast_add, Int.cast_pow] at hnn:ℤx:ℤy:ℤz:ℤhn:↑n = ↑x ^ 3 + ↑y ^ 3 + ↑z ^ 3⊢ ¬(↑n = ↑4 ∨ ↑n = ↑5)
  generalize (n : ZMod 9) = n at *n✝:ℤx:ℤy:ℤz:ℤn:ZMod 9hn:n = ↑x ^ 3 + ↑y ^ 3 + ↑z ^ 3⊢ ¬(n = ↑4 ∨ n = ↑5)
  generalize (x : ZMod 9) = x at *n✝:ℤx✝:ℤy:ℤz:ℤn:ZMod 9x:ZMod 9hn:n = x ^ 3 + ↑y ^ 3 + ↑z ^ 3⊢ ¬(n = ↑4 ∨ n = ↑5)
  generalize (y : ZMod 9) = y at *n✝:ℤx✝:ℤy✝:ℤz:ℤn:ZMod 9x:ZMod 9y:ZMod 9hn:n = x ^ 3 + y ^ 3 + ↑z ^ 3⊢ ¬(n = ↑4 ∨ n = ↑5)
  generalize (z : ZMod 9) = z at *n✝:ℤx✝:ℤy✝:ℤz✝:ℤn:ZMod 9x:ZMod 9y:ZMod 9z:ZMod 9hn:n = x ^ 3 + y ^ 3 + z ^ 3⊢ ¬(n = ↑4 ∨ n = ↑5)
  decide +revertAll goals completed! 🐙

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 := byr:ℚ⊢ IsSumOfThreeCubes r
  by_cases h : r = 0posr:ℚh:r = 0⊢ IsSumOfThreeCubes rnegr:ℚh:¬r = 0⊢ IsSumOfThreeCubes r
  ·posr:ℚh:r = 0⊢ IsSumOfThreeCubes r exact ⟨0, 0, 0, byr:ℚh:r = 0⊢ r = 0 ^ 3 + 0 ^ 3 + 0 ^ 3 norm_numr:ℚh:r = 0⊢ r = 0; exact hAll goals completed! 🐙⟩
  ·negr:ℚh:¬r = 0⊢ IsSumOfThreeCubes r let x := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)negr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)⊢ IsSumOfThreeCubes r
    let y := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2negr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)y:ℚ := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2⊢ IsSumOfThreeCubes r
    let z := (r ^ 2 + 6 * r + 3) * (- r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))negr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)y:ℚ := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2z:ℚ := (r ^ 2 + 6 * r + 3) * (-r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))⊢ IsSumOfThreeCubes r
    use x, y, zhr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)y:ℚ := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2z:ℚ := (r ^ 2 + 6 * r + 3) * (-r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))⊢ r = x ^ 3 + y ^ 3 + z ^ 3
    simp only [x, y, z]hr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)y:ℚ := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2z:ℚ := (r ^ 2 + 6 * r + 3) * (-r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))⊢ r =
  ((r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)) ^ 3 +
      ((3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2) ^ 3 +
    ((r ^ 2 + 6 * r + 3) * (-r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))) ^ 3
    field_simphr:ℚh:¬r = 0x:ℚ := (r ^ 6 + 45 * r ^ 4 - 81 * r ^ 2 + 27) / (6 * r * (r ^ 2 + 3) ^ 2)y:ℚ := (3 - r ^ 2) * (6 * r) / (r ^ 2 + 3) ^ 2z:ℚ := (r ^ 2 + 6 * r + 3) * (-r ^ 2 + 6 * r - 3) / (6 * r * (r ^ 2 + 3))⊢ r ^ 4 * 6 ^ 3 * (r ^ 2 + 3) ^ 6 =
  (r ^ 2 * (r ^ 2 * (r ^ 2 + 45) - 81) + 27) ^ 3 + r ^ 6 * 6 ^ 6 * (3 - r ^ 2) ^ 3 +
    (r ^ 2 + 3) ^ 3 * (r * (r + 6) + 3) ^ 3 * (r * (-r + 6) - 3) ^ 3
    ringAll goals completed! 🐙

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 declaration uses 'sorry'isSumOfThreeCubes_iff_mod_9 :
    answer(sorry) ↔ ∀ n : ℤ, IsSumOfThreeCubes n ↔ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9]) := by⊢ True ↔ ∀ (n : ℤ), IsSumOfThreeCubes n ↔ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9])
  sorryAll goals completed! 🐙


end SumOfThreeCubes