A Sieve on Rational Imbalances and the First Appearance of Denominators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911269557108736 |
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| author | Bilokon, Paul Alexander |
| author_facet | Bilokon, Paul Alexander |
| contents | We construct a sieve that enumerates rational ``imbalances'' of the form $(p-q)/(p+q)$ for integers $p\ge2$ and $1\le q<p$, ordered lexicographically by $(p,q)$. Each imbalance is reduced to lowest terms, and we record the sequence of distinct denominators as they first appear.
We show that every positive integer occurs exactly once as such a denominator, and that its first appearance coincides with the unit fraction $1/d$.
We then prove that the sieve, when viewed as a map from pairs $(p,q)$ to reduced fractions, enumerates all rational numbers in $(-1,1)$ without repetition, extend it symmetrically to all of $\mathbb{Q}$, and discuss its connections to hyperbolic geometry and rational enumeration theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Sieve on Rational Imbalances and the First Appearance of Denominators Bilokon, Paul Alexander General Mathematics 11Axx, 05Axx, 11Bxx G.2.1 We construct a sieve that enumerates rational ``imbalances'' of the form $(p-q)/(p+q)$ for integers $p\ge2$ and $1\le q<p$, ordered lexicographically by $(p,q)$. Each imbalance is reduced to lowest terms, and we record the sequence of distinct denominators as they first appear. We show that every positive integer occurs exactly once as such a denominator, and that its first appearance coincides with the unit fraction $1/d$. We then prove that the sieve, when viewed as a map from pairs $(p,q)$ to reduced fractions, enumerates all rational numbers in $(-1,1)$ without repetition, extend it symmetrically to all of $\mathbb{Q}$, and discuss its connections to hyperbolic geometry and rational enumeration theory. |
| title | A Sieve on Rational Imbalances and the First Appearance of Denominators |
| topic | General Mathematics 11Axx, 05Axx, 11Bxx G.2.1 |
| url | https://arxiv.org/abs/2510.22635 |