The Diophantine equation xy=z^n; for n=2,3,4,5,6; the Diophantine equation xyz=w^2; and the Diophantine system: xy=v^2 and yz=w^2

TL;DR

This paper completely characterizes positive integer solutions for specific Diophantine equations and systems involving three to five variables, providing parameterized solution families for each case.

Contribution

It offers explicit parameterized solutions for multiple Diophantine equations and systems, extending previous work by covering new cases and providing comprehensive solution descriptions.

Findings

Parameterized solutions for xy=z^n for n=2 to 6

Complete solutions for xyz=w^2

Solutions for the system xy=v^2 and yz=w^2

Abstract

In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for n=3, a 5-parameter of solutions; likewise for n=4. For n=5, a 7-parameter family of solutions; and likewise for n=6. See Theorems 2 through 6 respectively. The second goal of this paper, is determining all the positive integer solutions of xyz=w^2. This is done in Theorem7; the solution set is described in terms of six independent parameters. Finally, in Theorem 8, we achieve our third goal: determining all the positive integer solutions of the 5-variable Diophantine system: xy=v^2 and yz=w^2. The solution set is expressed in terms of eight parameters. This paper contains a total of four references.