On Properly $θ$-Congruent Numbers Over Real Number Fields
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918254840119296 |
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| author | Salami, Sajad Zargar, Arman Shamsi |
| author_facet | Salami, Sajad Zargar, Arman Shamsi |
| contents | The notion of $θ$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $θ$-congruent if it is the area of a rational triangle with an angle $θ$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $θ$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $θ$-congruent and properly $θ$-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to $6$, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree~$6$, and examine the exceptional cases $n=1, 2, 3$ and $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Properly $θ$-Congruent Numbers Over Real Number Fields Salami, Sajad Zargar, Arman Shamsi Number Theory primary 11G05, secondary 11R21, 11R16 The notion of $θ$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $θ$-congruent if it is the area of a rational triangle with an angle $θ$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $θ$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $θ$-congruent and properly $θ$-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to $6$, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree~$6$, and examine the exceptional cases $n=1, 2, 3$ and $6$. |
| title | On Properly $θ$-Congruent Numbers Over Real Number Fields |
| topic | Number Theory primary 11G05, secondary 11R21, 11R16 |
| url | https://arxiv.org/abs/2512.16597 |