Friday, September 21st

Problem

Find all positive integers $(x,y)$ such that $x^3+y$ and $x+y^3$ are divisible by $x^2+y^2$.


Solution

Consider $x^3+y-x(x^2+y^2)=y-xy^2=y(1-xy)$. Notice that in order to $x^3+y$ and $x+y^3$ being divisible by $x^2+y^2$, $gcd(x,y)=1$ as otherwise there is a contradiction. Hence $x^2+y^2$ divides $1-xy$, but $|1-xy|\leq x^2+y^2$, thus $x=y=1$.