Faithful Decomposition of Rationals

TL;DR

This paper introduces the concept of faithful decomposition of irreducible rationals, establishing lower bounds on the length of such decompositions and providing explicit constructions involving unit fractions.

Contribution

It defines faithful decomposition for rationals, proves lower bounds on their lengths, and constructs specific decompositions with mostly unit fractions.

Findings

Lower bound of t+2 on the length of faithful decompositions for certain fractions

Existence of faithful decompositions with only unit fractions plus one term

Explicit decomposition of 4/n into at most three fractions with minimal non-unit fractions

Abstract

If an irreducible fraction $\frac mn>0$ can be decomposed into the sum of several irreducible proper fractions with different denominators, and the positive number smaller than $\frac mn$ in fractional ideal $\frac 1n\mathbb Z$ 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 $\frac mn$ with $2\le t\le\frac mn<t+1$ 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 $\frac 4n$ as a faithful decomposition with three fractions at most one non-unit fraction.