A Note on Fraction Decompositions of Integers

Abstract

A decomposition of an integer n as a sum of several fractions is called faithful if no positive integer less than n can be obtained by replacing some of the numerators with smaller nonnegative integers. In this note, we prove the existence of faithful decompositions of all positive integers n, and determine the minimum number of fractions of the faithful decomposition. Here we also illustrate how the results we establish can be naturally translated into the language of Puiseux monoids.

Keywords:

• MSC: Primary 11D68
• Secondary 05A17
• 11P81
• 20M13

Acknowledgments

The author would like to thank his advisor, Chunwei Song, and all the anonymous referees for helpful comments.