Erdős Problem 282

formal-conjectures · erdos-problems · ams-5

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 $A$.

Does this process always terminate if $x$ has odd denominator and $A$ is the
set of odd numbers?

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/282
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/282.lean
Local target: Corpus.Erdos282.erdos_282
Source SHA-256: d6143e71d3f3a8e5be8d2668a8dc10143797e97cb8e9aa00c4499e25a75195f3
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record