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arXiv · 2502.19198

Faithful Decomposition of Rationals

Sunben Chiu·Pingzhi Yuan·Hongjian Li
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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.

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Sunben Chiu, Pingzhi Yuan, Hongjian Li. 2025-02-26. Faithful Decomposition of Rationals. https://arxiv.org/abs/2502.19198

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