2013/12/17 by M. A. Reynya, Reynya, M. A.
Mathematics · Physics and Astronomy · #11D25 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D25
paper · pdf · doi:10.48550/arxiv.1312.5702
4pages
arxiv created 2013/12/17 · openalex publication_date 2013/12/17 · arxiv updated 2013/12/20 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.