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

摘要

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