FormalConjectures.Wikipedia.Hall

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import FormalConjecturesUtil

Hall's conjecture

There exists a positive number $C$ such that for any integer $x, y$ with $y^2 \ne x^3$, $|y^2 - x^3| > C \sqrt{|x|}$.

References:

Wikipedia

L. Danilov, The Diophantine equation $x^3 - y^2 = k$ and Hall's conjecture, Mathematical notes of the Academy of Sciences of the USSR 32 (1982): 617-618

open Realnamespace Hall

def HallIneq (C : ℝ) (e : ℝ) : Prop :=
  ∀ x y : ℤ, y ^ 2 ≠ x ^ 3 → |y ^ 2 - x ^ 3| > C * (|x| : ℝ) ^ e

def HallConjectureExp (e : ℝ) : Prop := ∃ C : ℝ, C > 0 ∧ HallIneq C e

Original Hall's conjecture with exponent $1/2$.

@[category research open, AMS 11]
theorem hall_conjecture : HallConjectureExp 2⁻¹ := by⊢ HallConjectureExp 2⁻¹
  sorryAll goals completed! 🐙

Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be less than $0.0215$. Note that simple linarith does not work here.


@[category test, AMS 11]
theorem elkies_bound (C : ℝ) : HallIneq C 2⁻¹ → C < 0.0215 := byC:ℝ⊢ HallIneq C 2⁻¹ → C < 215e-4
  intro hC:ℝh:HallIneq C 2⁻¹⊢ C < 215e-4
  by_cases hC : C ≤ 0posC:ℝh:HallIneq C 2⁻¹hC:C ≤ 0⊢ C < 215e-4negC:ℝh:HallIneq C 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4
  ·posC:ℝh:HallIneq C 2⁻¹hC:C ≤ 0⊢ C < 215e-4 linarithAll goals completed! 🐙
  ·negC:ℝh:HallIneq C 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4 rw [HallIneqnegC:ℝh:∀ (x y : ℤ), y ^ 2 ≠ x ^ 3 → ↑|y ^ 2 - x ^ 3| > C * |↑x| ^ 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4negC:ℝh:∀ (x y : ℤ), y ^ 2 ≠ x ^ 3 → ↑|y ^ 2 - x ^ 3| > C * |↑x| ^ 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4] at hnegC:ℝh:∀ (x y : ℤ), y ^ 2 ≠ x ^ 3 → ↑|y ^ 2 - x ^ 3| > C * |↑x| ^ 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4
    specialize h 5853886516781223 447884928428402042307918negC:ℝh:447884928428402042307918 ^ 2 ≠ 5853886516781223 ^ 3 →
  ↑|447884928428402042307918 ^ 2 - 5853886516781223 ^ 3| > C * |↑5853886516781223| ^ 2⁻¹hC:¬C ≤ 0⊢ C < 215e-4
    simp at hnegC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ C < 215e-4
    have h1 : 76510695 < (5853886516781223 : ℝ) ^ (2 : ℝ)⁻¹ := byC:ℝ⊢ HallIneq C 2⁻¹ → C < 215e-4negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4
      norm_numC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ 76510695 < 5853886516781223 ^ (1 / 2)negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4
      rw [← sqrt_eq_rpowC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ 76510695 < √5853886516781223C:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ 76510695 < √5853886516781223negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4]C:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ 76510695 < √5853886516781223negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4
      refine lt_sqrt_of_sq_lt ?_C:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843⊢ 76510695 ^ 2 < 5853886516781223negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4
      norm_numnegC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹⊢ C < 215e-4
    have h2 : C * 76510695 < 1641843 := byC:ℝ⊢ HallIneq C 2⁻¹ → C < 215e-4negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹h2:C * 76510695 < 1641843⊢ C < 215e-4
      nlinarithnegC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹h2:C * 76510695 < 1641843⊢ C < 215e-4negC:ℝhC:¬C ≤ 0h:C * 5853886516781223 ^ 2⁻¹ < 1641843h1:76510695 < 5853886516781223 ^ 2⁻¹h2:C * 76510695 < 1641843⊢ C < 215e-4
    linarithAll goals completed! 🐙

Danilov proved that one cannot replace the exponent $1/2$ with larger number. In other words, for any $\delta > 0$, there is no positive constant $C$ such that $|y^2 - x^3| > C |x| ^ {1/2 + \delta}$ for all integers $x, y$ with $y^2 \ne x^3$.

@[category research solved, AMS 11]
theorem danilov (δ : ℝ) (h : δ > 0) : ¬ HallConjectureExp (2⁻¹ + δ) := byδ:ℝh:δ > 0⊢ ¬HallConjectureExp (2⁻¹ + δ) sorryAll goals completed! 🐙

Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \varepsilon$.

@[category research open, AMS 11]
theorem weak_hall_conjecture (ε : ℝ) (hε : ε > 0) : HallConjectureExp (2⁻¹ - ε) := byε:ℝhε:ε > 0⊢ HallConjectureExp (2⁻¹ - ε)
  sorryAll goals completed! 🐙

end Hall