Hall.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 3 contain sorry, 0 declared as axioms

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

Imports (1)

FormalConjecturesUtil

def Hall.HallIneq line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Int, Ne, HPow.hPow, Nat, OfNat.ofNat, GT.gt, abs, HSub.hSub, HMul.hMul
17 instances, coercions and auxiliary constants left out.
def Hall.HallConjectureExp line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Exists, And, GT.gt, OfNat.ofNat, Hall.HallIneq
3 instances, coercions and auxiliary constants left out.
theorem Hall.hall_conjecture Original Hall's conjecture with exponent $1/2$ sorry line 47 source
Source documentation
Original Hall's conjecture with exponent $1/2$.

depends on sorry what this means
Statement uses: Hall.HallConjectureExp, Inv.inv, Real, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hall_conjecture : HallConjectureExp 2⁻¹ := by
…
```
Mentions: Hall.HallConjectureExp
theorem Hall.elkies_bound Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be less than $0.0215$ line 55 source
Source documentation
Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be less than $0.0215$. Note that simple linarith does not work here.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Hall.HallIneq, Inv.inv, OfNat.ofNat, Nat, LT.lt, OfScientific.ofScientific, Bool.true
Proof uses: dite, LE.le, lt_of_not_ge, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, HMul.hMul, Neg.neg, HSub.hSub, inferInstance, AddMonoidWithOne, Mathlib.Meta.NormNum.instAtLeastTwo, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, Ring, Int.negOfNat, HPow.hPow, Nat.rawCast and 105 more
152 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
theorem elkies_bound (C : ℝ) : HallIneq C 2⁻¹ → C < 0.0215 := by
…
```
Mentions: Hall.HallIneq
theorem Hall.danilov Danilov proved that one cannot replace the exponent $1/2$ with larger number sorry line 77 source
Source documentation
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 $y^2 \ne x^3$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Not, Hall.HallConjectureExp, HAdd.hAdd, Inv.inv, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem danilov (δ : ℝ) (h : δ > 0) : ¬ HallConjectureExp (2⁻¹ + δ) := by sorry
…
```
Mentions: Hall.HallConjectureExp
theorem Hall.weak_hall_conjecture Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \varepsilon$ sorry line 83 source
Source documentation
Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \varepsilon$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Hall.HallConjectureExp, HSub.hSub, Inv.inv, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem weak_hall_conjecture (ε : ℝ) (hε : ε > 0) : HallConjectureExp (2⁻¹ - ε) := by
…
```
Mentions: Hall.HallConjectureExp