242.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 242

References:

erdosproblems.com/242
[Si56] Sierpiński, W., Sur les décompositions de nombres rationnels en fractions primaires. Mathesis (1956), 16--32.

Imports (1)

FormalConjecturesUtil

theorem Erdos242.erdos_242 For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$ sorry line 40 source
Source documentation
For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Exists, And, LE.le, Eq, Rat, HDiv.hDiv, HAdd.hAdd
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_242 (n : ℕ) (hn : 2 < n) :
    ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧
      (4 / n : ℚ) = 1 / x + 1 / y + 1 / z := by
…
```
theorem Erdos242.erdos_242.variants.schinzel_generalization Schinzel conjectured (see [Si56]) the generalisation that, for any fixed sorry line 51 source
Source documentation
Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x < y < z$ such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Filter.Eventually, Exists, And, LE.le, Eq, Rat, HDiv.hDiv, HAdd.hAdd, Filter.atTop
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 erdos_242.variants.schinzel_generalization
    (a : ℕ) (ha : 0 < a) :
    ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧
      (a / n : ℚ) = 1 / x + 1 / y + 1 / z := by
…
```