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Statement

Theorem 2 (p. 1020): "Any positive rational number $a/b$ where $b$ is odd, $a/b$ reduced, can be written as a finite sum of proper, reduced fractions whose numerators are distinct elements of the arithmetic progression $r+sx$, and whose denominators are distinct elements of the arithmetic progression $u+vy$; provided $(u,v)=1$, $(r,s)=1$, $(v,b)=1$, $(v,s)=1$, and $(v,r)=1$."

Here $(\cdot,\cdot)$ is the greatest common divisor. The print does not state the ranges of $r,s,u,v,x,y$ in the theorem.

Source. W. A. Webb, Sums of rational numbers, Canad. J. Math. 17 (1965), 1019--1024, doi:10.4153/cjm-1965-096-3; Theorem 2 on p. 1020, proof on pp. 1020--1023.

Read depth. Claims checked: the statement and the closing remarks of p. 1024 were read clause by clause on the print. The proof was followed in outline, not checked.

Proof pointer

The case $v=1$ follows from Theorem 1 (p. 1020). For $v>1$ the proof has two parts. The first (pp. 1020--1022) subtracts two fractions of the required kind, built from the congruence systems (1) and (2) and the size conditions (3), so that the remainder is a positive reduced fraction whose denominator is $\equiv1\pmod v$. The second (pp. 1022--1023) writes that remainder as a sum of copies of $1/b$ with $b\equiv1\pmod v$ and splits each copy by an explicit two-term identity on p. 1023, with the parameter chosen by the congruence system (5) so that both denominators lie in $u+vy$, and by the inequalities (6) so that all numerators and denominators are distinct.

On p. 1024 the author states that the conditions $(r,s)=1$ and $(v,b)=1$ are necessary for the theorem 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, he says that $(v,r)=1$ may be replaced by $(v,r,u-s)=1$, by an argument not given in the paper.

Dependencies

• Theorem 1 for the case $v=1$.

Bears on

• Problem 282: the theorem is an existence result for positive reduced rationals with odd denominator, with summands that are proper reduced fractions with distinct numerators in $r+sx$; it is not a statement about unit fractions, and the paper says nothing about the greedy algorithm the problem asks about.