Faithful Decomposition of Rationals

Abstract

If an irreducible fraction $\frac{m}{n}>0$ can be decomposed into the sum of several irreducible proper fractions with different denominators, and the positive number smaller than $\frac{m}{n}$ in fractional ideal $\frac{1}{n}\mathbb{Z}$ can not be obtained by replacing some numerator with smaller non-negative integers, then the decomposition is said to be faithful. For $t\in \mathbb{Z}$, we prove that the length of faithful decomposition of an irreducible fraction $\frac{m}{n}$ with $2\le t\le \frac{m}{n}<t+1$ is at least $t+2$. In addition, we show a faithful decomposition of rationals consisting only of unit fractions except for one term. And we write $\frac{4}{n}$ as a faithful decomposition with three fractions at most one non-unit fraction.