FormalConjectures.ErdosProblems.«242»

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import FormalConjecturesUtil

Erdős Problem 242

References:

erdosproblems.com/242

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


open scoped Topology


namespace Erdos242

For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'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 := byn:ℕhn:2 < n⊢ ∃ x y z, 1 ≤ x ∧ x < y ∧ y < z ∧ 4 / ↑n = 1 / ↑x + 1 / ↑y + 1 / ↑z
  sorryAll goals completed! 🐙

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 $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

@[category research open, AMS 11]
theorem declaration uses 'sorry'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 := bya:ℕha:0 < a⊢ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y z, 1 ≤ x ∧ x < y ∧ y < z ∧ ↑a / ↑n = 1 / ↑x + 1 / ↑y + 1 / ↑z
  sorryAll goals completed! 🐙


end Erdos242