I am interested in the following question
Do there exists infinity many positive integers $y$ for which $$1+f(y)^2\mid y^4-1$$forces $f(y)=y$ where $f:\mathbb{N}\to \mathbb{N}$ is a function.
I tried factorizing $y^4-1$ as $(y^2+1)(y^2-1)$ and then tried to find a condition for which $y^2-1$ and $f(y)^2+1$ are co-prime. But this method failed because we don't know anything about $f(n)$. Any help will be highly appreciated.