Some Diophantine equations related to positive-rank elliptic curves

TL;DR

This paper investigates specific Diophantine equations connected to positive-rank elliptic curves, establishing conditions for infinitely many rational solutions and expanding understanding of their structure.

Contribution

It provides new criteria on rational numbers that guarantee infinitely many solutions to certain symmetric Diophantine equations via positive-rank elliptic curves.

Findings

Conditions for infinite solutions to x+y+z=a+b+c and xyz=abc.

Conditions for infinite solutions to x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3.

Construction of families of positive-rank elliptic curves.

Abstract

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3. These results rely on exhibiting families of positive-rank elliptic curves.