Webb: Sums of Rational Numbers

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corollary_p1023: Webb's unnumbered corollary that Theorem 2 holds for b odd or even when u is a primitive root of v.

theorem_1: Webb's theorem that if an infinite set S of positive integers contains infinitely many disjoint relatively prime pairs, every rational number is a finite sum of reduced fractions with distinct numerators in S and distinct denominators.

theorem_2: Webb's theorem that a positive reduced rational a/b with b odd is a finite sum of proper reduced fractions with distinct numerators in the progression r + sx and distinct denominators in the progression u + vy, provided (u,v) = (r,s) = (v,b) = (v,s) = (v,r) = 1.

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The copy read for this card is the publisher's PDF of the Canad. J. Math. 17 article, 6 pages (PDF p. n is printed p. 1018+n). That PDF prints no copyright line, only the page footer "Downloaded from https://www.cambridge.org/core. 21 Sep 2026 at 17:35:54, subject to the Cambridge Core terms of use."; the journal's article page on Cambridge Core shows "Copyright © Canadian Mathematical Society 1965" (DOI 10.4153/cjm-1965-096-3, read 2026-10-02), every other right reserved.

W. A. Webb, "Sums of Rational Numbers," Canadian Journal of Mathematics, 17, 1019-1024, 1965. https://doi.org/10.4153/cjm-1965-096-3

Overview

W. A. Webb, "Sums of Rational Numbers," Canadian Journal of Mathematics 17 (1965), 1019--1024, studies finite decompositions of rationals in which the numerators, or both the numerators and the denominators, are restricted. Section 1 (p. 1019) recalls the unit-fraction background as cited results, not proved here: Breusch and Stewart showed that every rational number with an odd denominator is a sum of distinct odd unit fractions; Van Albada and Van Lint extended this to show that every integer is a sum of unit fractions with denominators from an arithmetic progression; Graham showed that a positive rational $a/b$ is a finite sum of reciprocals of distinct elements of $r+sx$ if and only if $(b/(b,(r,s)),\,r/(r,s))=1$, and proved a partition theorem.

Restricted numerators. Theorem 1 (Section 2, p. 1019; proof pp. 1019--1020): if an infinite set $S$ of positive integers contains infinitely many disjoint pairs of relatively prime elements, then every rational number $a/b$ is a finite sum of reduced fractions whose numerators are distinct elements of $S$ and whose denominators are distinct. The proof splits each copy of $1/b$ as

$$\frac1b=\frac{s_1}{P(s_1Q+s_2P)}+\frac{s_2}{Q(s_1Q+s_2P)},\qquad b=PQ,$$

with $(s_1,P)=(s_2,Q)=1$, taking successive pairs with rapidly growing sums so that all denominators are distinct. Webb notes on p. 1020 that the primes, the $k$th powers of the primes, any arithmetic progression $r+sx$ with $(r,s)=1$, and the Fibonacci numbers satisfy the hypothesis.

Restricted numerators and denominators. Theorem 2 (Section 3, p. 1020; proof pp. 1020--1023): a positive reduced rational $a/b$ with $b$ odd is a finite sum of proper reduced fractions whose numerators are distinct elements of $r+sx$ and whose denominators are distinct elements of $u+vy$, provided

$$(u,v)=(r,s)=(v,b)=(v,s)=(v,r)=1.$$

The print does not state the ranges of $x$ and $y$. The case $v=1$ follows from Theorem 1. For $v>1$ the first part of the proof (pp. 1020--1022) uses the congruence systems (1) and (2) and the size conditions (3) to write

$$\frac ab=\frac{r+x_1s}{U_1}+\frac{r+x_2s}{U_2}+\frac{a'}{b'},$$

with $U_1,U_2$ in $u+vy$, $a'/b'>0$ reduced and $b'\equiv1\pmod v$. The second part (pp. 1022--1023) splits each copy of $1/b$, for $b\equiv1\pmod v$, by the unnumbered identity on p. 1023,

$$\frac1b= \frac{r+z's}{b\{r+z's+b(r+(z'+1)s)\}} + \frac{r+(z'+1)s}{r+z's+b(r+(z'+1)s)},$$

with $z'$ chosen through the congruence system (5) so that both denominators lie in $u+vy$, and through the inequalities (6) so that all numerators and denominators are distinct.

The unnumbered corollary (p. 1023; proof pp. 1023--1024) states that Theorem 2 holds for $b$ odd or even if $u$ is a primitive root of $v$. The closing paragraph (p. 1024) says that $(r,s)=1$ and $(v,b)=1$ are necessary for Theorem 2 to hold in this generality, that $(u,v)=1$ appears almost impossible to omit, and that $(v,s)=1$ and $(v,r)=1$ may possibly be weakened; as an instance it says, without proof, that $(v,r)=1$ may be replaced by $(v,r,u-s)=1$.

Read status: claims checked for Theorems 1 and 2, the corollary and the remarks of pp. 1019, 1020 and 1024, read clause by clause on the print; the proofs were followed in outline, not checked. Result pages: theorem_1, theorem_2 and corollary_p1023.

Bears on. #282: the Introduction (p. 1019) recalls, as a cited result, the Breusch--Stewart theorem that every rational number with an odd denominator is a sum of distinct odd unit fractions; [[unit_fractions/webb_1965_sums_rational_numbers/theorem_2|Theorem 2]] (p. 1020) is an existence result for positive reduced rationals with odd denominator whose summands are proper reduced fractions with distinct numerators in $r+sx$; it is not a statement about unit fractions. The paper says nothing about the greedy algorithm.

Results.

• Theorem 1 (p. 1019): with numerators from an infinite set containing infinitely many disjoint coprime pairs, every rational is a finite sum of reduced fractions with distinct numerators and distinct denominators.
• Theorem 2 (p. 1020): a positive reduced rational with odd denominator is a finite sum of proper reduced fractions with distinct numerators in $r+sx$ and distinct denominators in $u+vy$, under $(u,v)=(r,s)=(v,b)=(v,s)=(v,r)=1$.
• Corollary (p. 1023): Theorem 2 holds for $b$ odd or even if $u$ is a primitive root of $v$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.