Does there exist a non-constant function $f:\mathbb N^2 \rightarrow \mathbb N$ such that $f(x,y)+f(y,x)=f(x^2,y^2)+1$

Question

Does there exist a non-constant function $f:\mathbb N^2 \rightarrow \mathbb N$ such that $$f(x,y)+f(y,x)=f(x^2,y^2)+1$$ for all positive integers $x,y$?

I think that such a function does not exist. But I do not know how to prove

Answer 1

How about this function?

$$ f(x, y) = \begin{cases} 2^n + 1 & \text{if } (x, y) = (2^{2^n}, 1) \text{ or } (1, 2^{2^n}) \\ 1 & \text{otherwise} \end{cases} $$