On generalised Pythagorean triples over number fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913891837018112 |
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| author | García, Pedro-José Cazorla |
| author_facet | García, Pedro-José Cazorla |
| contents | Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular, we can now determine whether $E_{a,b,c}$ has solutions and find them in a computationally effective manner.
In this paper, we consider an extension of generalised Pythagorean triples to number fields $K$. In particular, we survey and extend the existing results over $\mathbb{Q}$ for determining if $E_{a,b,c}$ has solutions over number fields and if so, to find and parameterise them, as well as to find a minimal solution. Throughout the text, we incorporate numerous examples to make our results accessible to all researchers interested in the topic of generalised Pythagorean triples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalised Pythagorean triples over number fields García, Pedro-José Cazorla Number Theory Primary 11D09, Secondary 11R04, 11Y50, 14G12 Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular, we can now determine whether $E_{a,b,c}$ has solutions and find them in a computationally effective manner. In this paper, we consider an extension of generalised Pythagorean triples to number fields $K$. In particular, we survey and extend the existing results over $\mathbb{Q}$ for determining if $E_{a,b,c}$ has solutions over number fields and if so, to find and parameterise them, as well as to find a minimal solution. Throughout the text, we incorporate numerous examples to make our results accessible to all researchers interested in the topic of generalised Pythagorean triples. |
| title | On generalised Pythagorean triples over number fields |
| topic | Number Theory Primary 11D09, Secondary 11R04, 11Y50, 14G12 |
| url | https://arxiv.org/abs/2506.11636 |