On the Diophantine equation $f(x)=2f(y)$

TL;DR

This paper proves that the specific polynomial equation $f(x)=2f(y)$ has no positive integer solutions except for the trivial case where both variables are 1, highlighting a unique solution property of this Diophantine equation.

Contribution

It establishes the non-existence of solutions for the equation $f(x)=2f(y)$ in positive integers, except for the trivial solution, for a particular polynomial function.

Findings

No solutions in positive integers except (1,1)

Unique solution at the trivial point

Advances understanding of polynomial Diophantine equations

Abstract

Let $f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3).$ We prove that the Diophantine equation $ f(x)=2f(y)$ has no solutions in positive integers $x$ and $y$, except $(x, y)=(1, 1)$.