Erdős Problem #282

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The odd-denominator condition refers to the rational number in lowest terms. At each stage the current positive remainder determines the next allowed denominator; reaching remainder zero terminates the algorithm. The set of permitted denominators is fixed throughout the expansion. [S1] [S2]

Statement recorded in the supplied source.

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? More generally, for which pairs $x$ and $A$ does this process terminate? [S1]

Short English statement

When does the greedy reciprocal expansion terminate if only specified denominators are allowed, particularly odd denominators for rationals with odd reduced denominator?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2029 (EP-282), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 282, https://www.erdosproblems.com/282. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 R. L. Graham, On Finite Sums of Unit Fractions, Proceedings of the London Mathematical Society s3-14 (1964), 193-207,

  \nolinkurl{doi:10.1112/plms/s3-14.2.193.}

  https://doi.org/10.1112/plms/s3-14.2.193. Bibliographic information from the supplied record.

• S4 R. L. Graham, On Finite Sums of Reciprocals of Distinct nth Powers, Pacific Journal of Mathematics 14 (1964), 85-92. https://msp.org/pjm/1964/14-1/pjm-v14-n1-p10-p.pdf. Bibliographic information from the supplied record.

• S5 Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) (1964), 193-207. Bibliographic information from the supplied record.