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arXiv · 2403.19694

The diophantine equation $x^4+y^4=z^4+w^4$

Ajai Choudhry·Arman Shamsi Zargar
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Abstract

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.

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Ajai Choudhry, Arman Shamsi Zargar. 2024-03-18. The diophantine equation $x^4+y^4=z^4+w^4$. https://arxiv.org/abs/2403.19694

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