680.lean`

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L4DEL: Lean4 Domain Explanation Language

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 680

Reference: erdosproblems.com/680

Imports (1)

FormalConjecturesUtil

theorem Erdos680.erdos_680.parts.i Is it true that, for all sufficiently large $n$, there exists some $k$ such that $$ p(n+k)>k^2+1, $$ where $p(m)$ denotes the least prime factor of $m$? sorry line 41 source
Source documentation
Is it true that, for all sufficiently large $n$, there exists some $k$ such that $$ p(n+k)>k^2+1, $$ where $p(m)$ denotes the least prime factor of $m$?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Exists, And, Ne, OfNat.ofNat, GT.gt, Nat.minFac, HAdd.hAdd, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_680.parts.i :
    answer(sorry) ↔ ∀ᶠ (n : ℕ) in .atTop, ∃ k ≠ 0, (n + k).minFac > k^2 + 1 := by
…
```
theorem Erdos680.erdos_680.parts.ii Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant? sorry line 50 source
Source documentation
Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Exists, And, Not, Filter.Eventually, Nat, Ne, Nat.minFac, HAdd.hAdd, Real.exp, HMul.hMul, Real.sqrt, 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_680.parts.ii : answer(sorry) ↔ ∀ ε > 0, ∃ C > 0,
    ¬ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k ≠ 0,
    Nat.minFac (n + k) > exp ((1 + ε) * √k) + C := by
…
```