Faithful Decomposition of Rationals
Sunben Chiu, Pingzhi Yuan, Hongjian Li
TL;DR
This work studies faithful decompositions of positive rationals relative to $\frac{1}{n}\mathbb{Z}$ and establishes a sharp lower bound on decomposition length: for $2\le t\le \frac{m}{n}<t+1$, any faithful decomposition of $\frac{m}{n}$ has length at least $t+2$, with infinitely many achieving equality. It develops general structural and coprimality properties for faithful decompositions, proves a key equality $S_e=T_e$ that enables systematic construction of decompositions with many unit-fraction terms, and provides explicit constructions for decompositions of $\frac{4}{n}$ into three fractions with at most one non-unit term. The paper also delivers explicit faithful decompositions of $\frac{4}{n}$ depending on $n$ modulo $4$, leveraging Diophantine and arithmetic-progression techniques and addressing exceptional cases. Collectively, the results advance constructive understanding of Egyptian-like faithful decompositions and offer broad methods for assembling such decompositions in rational arithmetic.
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.