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arXiv · 1312.5702

Complete description of rational points of Diophantine equation x4+y4=z4+w4

M. A. Reynya
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Abstract

In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operation on the cubic curves. (3)We present an algorithm to find all rational points on our Quartica.

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M. A. Reynya. 2013-12-17. Complete description of rational points of Diophantine equation x4+y4=z4+w4. https://arxiv.org/abs/1312.5702

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