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Statement

Corollary (p. 1023, unnumbered): "The above theorem holds for $b$ odd or even if $u$ is a primitive root of $v$."

That is, under the hypotheses of Theorem 2 other than the oddness of $b$, namely $(u,v)=(r,s)=(v,b)=(v,s)=(v,r)=1$, and when $u$ is a primitive root of $v$, every positive reduced rational $a/b$ is a finite sum of proper reduced fractions with distinct numerators in $r+sx$ and distinct denominators in $u+vy$.

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

Read depth. Claims checked: the statement was read on the print. The proof was followed in outline, not checked.

Proof pointer

The proof (pp. 1023--1024) subtracts fractions with numerators in $r+sx$ and denominators in $u+vy$ one at a time, using that $u$ is a primitive root of $v$ to choose the number of steps $t$ with $bu^t\equiv1\pmod v$, so that the positive remainder has denominator $\equiv1\pmod v$; the second part of the proof of Theorem 2 then finishes.

Dependencies

• Theorem 2, second part of its proof.

Bears on

No Erdős problem in the corpus.