Find all positive integers $x,y$ such that $(x+y)(xy+1)$ is a power of $2$
It is clear that as we are given a factorisation of the power of two, both of those terms have to be powers of two. $x+y$ is then even and $xy$ is odd, which only happens when both $x,y$ are odd.
Then some obvious solutions include $x,y = \left\{ 1,1 \right\},\left\{ 1,3 \right\},\left\{ 1,7 \right\},...$
I am not sure how to proced with finding the complete set (and proving its completeness). Perhaps examination modulo some other integer? I have not been successful with $3$.
Thank you for your help.