368.lean`

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

9 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 368

References:

erdosproblems.com/368
[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (1976), 25–44.
[Po18] Pólya, Georg, Zur arithmetischen Untersuchung der Polynome. Math. Z. (1918), 143–148.
[Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177–236.

Imports (1)

FormalConjecturesUtil

def Erdos368.F The largest prime factor of $n(n+1)$ line 41 source
Source documentation
The largest prime factor of $n(n+1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.maxPrimeFac, HMul.hMul, HAdd.hAdd, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def F (n : ℕ) : ℕ := (n * (n + 1)).maxPrimeFac
…
```
theorem Erdos368.erdos_368.lower_bound How large is the largest prime factor of $n(n+1)$? sorry line 48 source
Source documentation
How large is the largest prime factor of $n(n+1)$? The truth is probably $F(n)\gg (\log n)^2$ for all $n$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, HMul.hMul, HPow.hPow, Real.log, Erdos368.F
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_368.lower_bound :
    ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 1 ≤ n → c * Real.log n ^ 2 ≤ (F n : ℝ) := by
…
```
Mentions: c, Erdos368.F
theorem Erdos368.erdos_368.upper_bound Erdős [Er76d] conjectured that, for every $\epsilon>0$, there are infinitely many $n$ such that $F(n)<(\log n)^{2+\epsilon}$ sorry line 57 source
Source documentation
Erdős [Er76d] conjectured that, for every $\epsilon>0$, there are infinitely many $n$ such that $F(n)<(\log n)^{2+\epsilon}$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Frequently, Nat, Erdos368.F, HPow.hPow, Real.log, HAdd.hAdd, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_368.upper_bound :
    ∀ ε : ℝ, 0 < ε → ∃ᶠ n : ℕ in atTop, (F n : ℝ) < Real.log n ^ (2 + ε) := by
…
```
Mentions: Erdos368.F
theorem Erdos368.erdos_368.variants.tendsto Pólya [Po18] proved that $F(n)\to\infty$ as $n\to\infty$ sorry line 63 source
Source documentation
Pólya [Po18] proved that $F(n)\to\infty$ as $n\to\infty$.

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Erdos368.F, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_368.variants.tendsto : Tendsto F atTop atTop := by
…
```
Mentions: Erdos368.F
theorem Erdos368.erdos_368.variants.subpower For every $\delta>0$, there are infinitely many $n$ with $F(n)\leq n^\delta$ sorry line 72 source
Source documentation
For every $\delta>0$, there are infinitely many $n$ with $F(n)\leq n^\delta$. This follows from Schinzel's [Sc67b] observation that, for infinitely many $n$, $F(n)\leq n^{O(1/\log\log\log n)}$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Frequently, Nat, LE.le, Erdos368.F, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_368.variants.subpower :
    ∀ δ : ℝ, 0 < δ → ∃ᶠ n : ℕ in atTop, (F n : ℝ) ≤ (n : ℝ) ^ δ := by
…
```
Mentions: Erdos368.F
example Erdos368.(example at line 77) line 77 source
example Erdos368.(example at line 80) line 80 source
example Erdos368.(example at line 83) line 83 source
example Erdos368.(example at line 86) line 86 source