445.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 445

References:

erdosproblems.com/445
[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.
MathOverflow

Imports (1)

FormalConjecturesUtil

def Erdos445.Erdos445Prop The property that there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$ line 40 source
Source documentation
The property that there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: Exists, And, LT.lt, HAdd.hAdd, HPow.hPow, Nat.ModEq, HMul.hMul, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.
```
def Erdos445Prop (c : ℝ) (p n : ℕ) : Prop :=
…
```
Mentions: c
theorem Erdos445.erdos_445 Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$? sorry line 53 source
Source documentation
Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
This is discussed in this MathOverflow question [MathOverflow].

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, HDiv.hDiv, OfNat.ofNat, Nat, Filter.Eventually, Nat.Prime, Erdos445.Erdos445Prop, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_445 :
    answer(sorry) ↔ ∀ c : ℝ, c > 1 / 2 →
      ∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := by
…
```
Mentions: c, Erdos445.Erdos445Prop
theorem Erdos445.erdos_445.variants.heilbronn Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$ sorry line 62 source
Source documentation
Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, GT.gt, Filter.Eventually, Nat, Nat.Prime, Erdos445.Erdos445Prop, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_445.variants.heilbronn :
    ∃ c₀ < 1, ∀ c : ℝ, c > c₀ →
      ∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := by
…
```
Mentions: c, Erdos445.Erdos445Prop
theorem Erdos445.erdos_445.variants.heath_brown Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$ sorry line 71 source
Source documentation
Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$.

depends on sorry what this means
Statement uses: Real, GT.gt, HDiv.hDiv, OfNat.ofNat, Nat, Filter.Eventually, Nat.Prime, Erdos445.Erdos445Prop, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_445.variants.heath_brown :
    ∀ c : ℝ, c > 3 / 4 →
      ∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n := by
…
```
Mentions: c, Erdos445.Erdos445Prop
theorem Erdos445.erdos_445.test.small_example Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \in (0,5)$ satisfies $2 \cdot 3 = 6 \equiv 1 \pmod{5}$ line 79 source
Source documentation
Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \in (0,5)$ satisfies $2 \cdot 3 = 6 \equiv 1 \pmod{5}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos445.Erdos445Prop, OfNat.ofNat, Real, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, HAdd.hAdd, HPow.hPow, Nat.ModEq, HMul.hMul, And.intro, Decidable.byContradiction, Not, Eq.mpr, id, Eq, congrArg, Real.rpow_one, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 6 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_445.test.small_example : Erdos445Prop 1 5 1 := by
…
```
Mentions: Erdos445.Erdos445Prop