Louwsma–Martino: Rational Numbers with Two-Term Odd Greedy Expansion

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The retained folder-name PDF is the INTEGERS 25 (2025) #A46 article, 7 pages numbered 1–7; a Markdown reading copy sits beside it. The Zenodo record whose DOI the file prints on p. 1 names the license "Creative Commons Attribution 4.0 International" (https://zenodo.org/records/15536292, read 2026-10-02), the Creative Commons Attribution 4.0 license; the journal's site also states that all its works carry that license.

Joel Louwsma and Joseph Martino, "Rational Numbers with Two-Term Odd Greedy Expansion," Integers 25 (2025), #A46, 7 pp. https://doi.org/10.5281/zenodo.15536292

Overview

Joel Louwsma and Joseph Martino, “Rational Numbers with Two-Term Odd Greedy Expansion,” Integers 25 (2025), #A46, DOI 10.5281/zenodo.15536292, gives a complete elementary classification of positive rationals whose odd greedy expansion terminates after exactly two terms. The algorithm selects the largest unit fraction with odd denominator not exceeding the current remainder; the authors allow both $1/1$ and repeated terms (Section 1, p. 2). They recall, without proving, that unrestricted greedy expansions terminate, that rationals with odd reduced denominator possess some finite odd-unit-fraction representation [1, 9], and that termination of the greedy odd expansion for every such rational remains open (Section 1, pp. 1–2).

The preliminary parity result says that a sum of an even number of odd-denominator unit fractions has even numerator in every fractional representation (Proposition 1, pp. 2–3). Its proof writes the sum over the odd common denominator $x_1\cdots x_m$; the numerator is the degree-$m-1$ elementary symmetric polynomial, a sum of $m$ odd terms, and hence is even when $m$ is even.

For positive integers $n,d$ and an odd positive integer $x_1$, put $r=nx_1-d$. Lemma 2 (p. 3) proves that $x_1$ is the first greedy denominator for $n/d$ exactly when

$$0\le r<2n.$$

This is just the defining interval $1/x_1\le n/d<1/(x_1-2)$, with the paper’s separate convention for $x_1=1$.

The principal result is Theorem 3 (pp. 4–5). Let $n$ be even, choose $r$ with $0<r<2n$ and $v_2(2r)\le v_2(n)$, let $p_1,\ldots,p_s$ be the prime divisors of $r$, and define

$$a_i=\max\!\left\{\left\lceil\frac{v_{p_i}(2r)-v_{p_i}(n)}2\right\rceil,0\right\},\qquad P=\prod_{i=1}^s p_i^{a_i}.$$

Then the fractions with a two-term odd greedy expansion are precisely

$$\frac{n}{nP(1+2t)-r},\qquad t\ge0.$$

The proof is a divisibility classification rather than an asymptotic or computational argument. After the first term,

$$\frac nd-\frac1{x_1}=\frac r{dx_1}.$$

Thus the expansion has exactly two terms iff $0<r<2n$ and $dx_1/r$ is an odd integer. Since $d=nx_1-r$ and $x_1$ is odd, the latter condition is equivalent to $nx_1^2/(2r)\in\mathbb Z$. Comparing valuations gives the stated condition at $2$ and the lower bounds $v_{p_i}(x_1)\ge a_i$; consequently $x_1=P(1+2t)$ (Theorem 3, pp. 4–5). For fixed $n$, this places the admissible denominators in at most $2n-2$ arithmetic sequences (p. 5).

Corollary 5 (pp. 5–6) specializes the classification to reduced fractions. Since

$$\gcd\bigl(n,nP(1+2t)-r\bigr)=\gcd(n,r),$$

reducedness is equivalent to $\gcd(r,2n)=1$, and then $a_i=\lceil v_{p_i}(r)/2\rceil$. Hence the reduced two-term fractions are exactly

$$\frac{n}{n\left(\prod_{p\mid r}p^{\lceil v_p(r)/2\rceil}\right)(1+2t)-r},$$

where $n$ is even, $0<r<2n$, $\gcd(r,2n)=1$, and $t\ge0$. The authors note that this gives exactly $\phi(2n)$ denominator progressions for each fixed even reduced numerator $n$ (p. 6). Examples 4 and 6 work out the cases $n=4$ and $n=2$, respectively (pp. 5–6). The paper treats no expansions of length at least three and supplies neither a general termination theorem nor a nonterminating odd-denominator example.

Relation to E282

This source bears on Problem 282.

For E282, write the current nonzero remainder in lowest terms as $x=a/b$, with $b$ odd, and let

$$q_0=\min\{q\in\mathbb N:q\text{ odd and }q\ge b/a\},\qquad r=aq_0-b.$$

Lemma 2 translates exactly to $0\le r<2a$. If $r=0$, then $x=1/q_0$ and the process ends in one step. If $r>0$, its next remainder is

$$x-\frac1{q_0}=\frac{r}{bq_0}.$$

Theorem 3’s proof therefore gives the particularly useful local criterion

$$\text{the trajectory ends after exactly two steps}\quad\Longleftrightarrow\quad \frac{bq_0}{r}\text{ is an odd integer}.$$

Equivalently, the second denominator is $q_1=bq_0/r$. This can serve as an explicit terminal test at any stage of an E282 trajectory.

For an initial reduced $x=a/b\in(0,1)$, Corollary 5 gives the complete two-step terminal locus. It requires $a$ even and parameters

$$0<r<2a, \quad \gcd(r,2a)=1, \quad P(r)=\prod_{p\mid r}p^{\lceil v_p(r)/2\rceil}, \quad q_0=P(r)(1+2t),$$

with

$$b=aq_0-r>a.$$

Conversely, every such choice produces a reduced E282 input with odd $b$ whose greedy expansion has exactly two terms. Proposition 1 also shows that no reduced fraction with odd numerator can terminate in exactly two terms.

The result is thus usable as an explicit absorbing family: an attempted proof of E282 could try to show that every trajectory eventually reaches either an odd unit fraction or one of these two-step families. It also supplies exact divisibility and valuation conditions for computations or for excluding proposed two-step endings. It does not show that an arbitrary odd-denominator trajectory reaches this locus, control expansions of three or more terms, provide a decreasing invariant, or rule out an infinite trajectory; consequently it does not resolve E282.

The paper permits repeated denominators. This matches the literal recursive choice in E282, although E282’s statement also describes the resulting fractions as distinct. The conventions agree below $2/3$; within $(0,1)$, the relevant two-term exception is $2/3=1/3+1/3$ (Section 1, p. 2).