The diophantine equation $x^4+y^4=z^4+w^4$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913289382920192 |
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| author | Choudhry, Ajai Zargar, Arman Shamsi |
| author_facet | Choudhry, Ajai Zargar, Arman Shamsi |
| contents | Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$ respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The diophantine equation $x^4+y^4=z^4+w^4$ Choudhry, Ajai Zargar, Arman Shamsi General Mathematics 11D25 Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$ respectively. |
| title | The diophantine equation $x^4+y^4=z^4+w^4$ |
| topic | General Mathematics 11D25 |
| url | https://arxiv.org/abs/2403.19694 |