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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2502.19198 |
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| _version_ | 1866916631917101056 |
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| author | Chiu, Sunben Yuan, Pingzhi Li, Hongjian |
| author_facet | Chiu, Sunben Yuan, Pingzhi Li, Hongjian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Faithful Decomposition of Rationals Chiu, Sunben Yuan, Pingzhi Li, Hongjian Number Theory 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. |
| title | Faithful Decomposition of Rationals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.19198 |