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A NOTE ON THE SUM OF RECIPROCALS

Part of: Elementary number theory Sequences and sets Combinatorics

Published online by Cambridge University Press:  17 May 2019

YUCHEN DING  and
YU-CHEN SUN Open the ORCID record for YU-CHEN SUN [Opens in a new window]
YUCHEN DING
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China email [email protected]
YU-CHEN SUN*
Affiliation:
Medical School, Nanjing University, Nanjing 210093, People’s Republic of China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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We prove that, given a positive integer $m$, there is a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that

$$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$
with the property that partial sums of the series $\{1/n_{i}\}_{i=1}^{k}$ do not represent other integers.

Keywords

sum of reciprocalsEgyptian fractioncover of the integers

MSC classification

Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Secondary: 05A17: Partitions of integers 11A51: Factorization; primality 11B75: Other combinatorial number theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 189 - 193
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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