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Sums of Rational Numbers

Published online by Cambridge University Press:  20 November 2018

W. A. Webb
W. A. Webb*
Affiliation:
Michigan State University
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Previously, the problem of expressing rational numbers as finite sums of rational numbers of a given type has been concerned with the Egyptian, or unit, fractions. It has long been known that any rational number is the sum of distinct unit fractions. In response to a problem proposed by E. P. Starke (4), R. Breusch (1) and B. M. Stewart (5) showed that every rational number with an odd denominator is a sum of distinct odd unit fractions. P. J. Van Albada and J. H. Van Lint (6) extended this result to show that any integer is a sum of unit fractions with denominators from an arithmetic progression.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 1019 - 1024
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Breusch, R., A special case of Egyptian fractions, Amer. Math. Monthly, 61 (1954), 200–201.Google Scholar
2. Graham, R. L., On finite sums of unit fractions, Proc. London Math. Soc. (3), 14 (1964), 193–207.Google Scholar
3. Graham, R. L., A theorem of partitions, J. Austral. Math. Soc. III-4 (1963), 435–441.Google Scholar
4. Starke, E. P., Advanced problem 4512, Amer. Math. Monthly, 59 (1952), 640.Google Scholar
5. Stewart, B. M., Sums of distinct divisors, Amer. J. Math., 76 (1954), 779–785.Google Scholar
6. Van, P. J. Albada and J. H. Van Lint, Reciprocal bases for the integers, Amer. Math. Monthly, 70 (1963), 170–174.Google Scholar