Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Question

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations,

$x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?

Answer 1

We can prove that the map $f : (x,y) \mapsto (x^2+y^2,x^3+y^3)$, when restricted to the inputs $0 \le y \le x$ is injective. So the only way we can have both equalities is if $\{x,y\} = \{u,v\}$. And the result is valid even if you replace "positive integer" with "positive real".

Of course, if $x^2+y^2 = 0 $ then $x=y=0$ is the only possibility.
So let's suppose that $x^2+y^2$ and $x$ are strictly positive. Let $t = \frac yx$. Then $\frac{(x^3+y^3)^2}{(x^2+y^2)^3} = \frac{1+2t^3+t^6}{1+3t²+3t^4+t^6}$ is decreasing from $1$ at $t=0$ to $\frac 12$ at $t=1$. This shows that $t$ is determined by $f(x,y)$. Then once you have $t$, $x= \sqrt \frac {x^2+y^2}{1+t^2}$ and so on with $y$, which shows that $x$ and $y$ are determined by $f(x,y)$.