Does every rational number with an odd denominator have an odd greedy expansion

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write a positive rational number in lowest terms as $r_0=a/b$ with $a,b\in\mathbb N$ and $b$ odd. As long as $r_j>0$, choose the least positive odd integer $d_j$ with $1/d_j\le r_j$ and set $r_{j+1}=r_j-1/d_j$. The choice permits $d_j=1$ and repeated denominators. It is the largest available unit fraction not exceeding the remainder. The expansion terminates if some remainder becomes zero.

Statement recorded in the supplied source.

Does this odd greedy algorithm terminate for every such $a/b$, yielding $a/b=\sum_{j=0}^{m-1}1/d_j$ with finitely many terms? [S2]

Short English statement

Must the greedy algorithm restricted to odd denominators eventually express every positive rational with odd denominator as a finite sum of unit fractions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400095 (AMR-093-0095). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joel Louwsma and Joseph Martino, Rational numbers with odd greedy expansion of fixed length, arXiv:2309.07280v1 (2023) Introduction, pp. 1--2. https://arxiv.org/pdf/2309.07280.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Does every rational number with an odd denominator have an odd greedy expansion. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.