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Statement

Theorem 1 (p. 1019): "Given any infinite set $S$ of positive integers such that in $S$ there are infinitely many disjoint pairs of elements which are relatively prime, then any rational number $a/b$ may be written as a finite sum of reduced fractions whose numerators are distinct elements of $S$ and whose denominators are distinct."

The statement covers every rational number, not only positive ones; for a negative $a/b$ the proof makes all the denominators negative (p. 1019). The paper 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 are suitable sets $S$.

Source. W. A. Webb, Sums of rational numbers, Canad. J. Math. 17 (1965), 1019--1024, doi:10.4153/cjm-1965-096-3; Theorem 1 on p. 1019, proof on pp. 1019--1020.

Read depth. Claims checked: the statement and the remark on suitable sets were read clause by clause on the print (pp. 1019--1020). The proof was followed in outline, not checked.

Proof pointer

The proof (pp. 1019--1020) writes $a/b$ as a sum of copies of $1/b$ (of $1/2$ when $b=1$) and splits each copy into two fractions whose numerators form a coprime pair of $S$, after factoring $b$ so that each numerator is prime to the matching factor. Each new pair is taken with a sum large enough that every new denominator exceeds every earlier one, which keeps the numerators and the denominators distinct.

Dependencies

None outside the paper.

Bears on

No Erdős problem in the corpus.