Rational angle bisectors on the coordinate plane and solutions of Pell's equations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913630094622720 |
|---|---|
| author | Hirotsu, Takashi |
| author_facet | Hirotsu, Takashi |
| contents | On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions of this equation with solutions of negative Pell's equations was discovered by the author. In this article, for a given square-free integer $d > 1$ and a given integer $z > 1,$ we describe every integral solution $(x,y)$ of $|x^2-dy^2| = z$ such that $x$ and $dy$ are coprime by using the fundamental unit of $\mathbb Q(\sqrt d)$ and elements of $\mathbb Z[\sqrt d]$ whose absolute value of norms are the smallest prime powers. We also describe every nontrivial rational solution of the above equation as one of its applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01091 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational angle bisectors on the coordinate plane and solutions of Pell's equations Hirotsu, Takashi Number Theory 11D25, 11D09, 11D57 On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions of this equation with solutions of negative Pell's equations was discovered by the author. In this article, for a given square-free integer $d > 1$ and a given integer $z > 1,$ we describe every integral solution $(x,y)$ of $|x^2-dy^2| = z$ such that $x$ and $dy$ are coprime by using the fundamental unit of $\mathbb Q(\sqrt d)$ and elements of $\mathbb Z[\sqrt d]$ whose absolute value of norms are the smallest prime powers. We also describe every nontrivial rational solution of the above equation as one of its applications. |
| title | Rational angle bisectors on the coordinate plane and solutions of Pell's equations |
| topic | Number Theory 11D25, 11D09, 11D57 |
| url | https://arxiv.org/abs/2305.01091 |