Erdős problem 282 · problems.science

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$ : choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$ . If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$ .

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8 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

282.lean

Retained formal statement5 of 8

Graham has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$ . Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always.

FormalConjectures/ErdosProblems/282.leanErdos282.erdos_282.variants.sq4 linesExact file

1sorry ↔2  ∀ (x : ℚ),3    ↑x ∈ Set.Ico 0 (Real.pi ^ 2 / 6 - 1) ∪ Set.Ico 1 (Real.pi ^ 2 / 6) →4      Erdos282.greedyUnitFractionRem {n | IsSquare n} x =ᶠ[Filter.atTop] 0

OpenStatement only, no proof

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