688.lean`

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5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 688

Reference:

erdosproblems.com/688
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Imports (1)

FormalConjecturesUtil

def Erdos688.Erdos688Prop Define line 38 source
Source documentation
Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n} < p \leq n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p \pmod p$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Exists, LE.le, OfNat.ofNat, And, Nat.Prime, LT.lt, HPow.hPow, Nat.ModEq
7 instances, coercions and auxiliary constants left out.
```
def Erdos688Prop (n : ℕ) (ε : ℝ) : Prop :=
…
```
def Erdos688.epsilonFunction line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: SupSet.sSup, Set.ofPred, Erdos688.Erdos688Prop
1 instances, coercions and auxiliary constants left out.
theorem Erdos688.erdos_688.parts.i Estimate $\epsilon_n$ sorry line 49 source
Source documentation
Estimate $\epsilon_n$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos688.epsilonFunction, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_688.parts.i :
    epsilonFunction =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos688.epsilonFunction
theorem Erdos688.erdos_688.parts.ii In particular, is it true that $\epsilon_n = o(1)$? sorry line 57 source
Source documentation
In particular, is it true that $\epsilon_n = o(1)$?

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos688.epsilonFunction, OfNat.ofNat
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_688.parts.ii : answer(sorry) ↔
    epsilonFunction =o[atTop] (fun (n : ℕ) ↦ (1 : ℝ)) := by
…
```
Mentions: Erdos688.epsilonFunction
theorem Erdos688.erdos_688.variants.lglglg_over_lglg_is_big_o Erdős claims in [Er80] (p sorry line 66 source
Source documentation
Erdős claims in [Er80] (p. 106) that it is not difficult to prove $\epsilon_n \gg \frac{\log\log\log n}{\log\log n}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HDiv.hDiv, Real.log, Erdos688.epsilonFunction
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_688.variants.lglglg_over_lglg_is_big_o :
    (fun (n : ℕ) ↦ (log (log (log (n : ℝ)))) / (log (log (n : ℝ))))
      =O[atTop] epsilonFunction := by
…
```
Mentions: Erdos688.epsilonFunction