Is there any hope of proving the following conjecture?
For any integer $x\geq1007$, there exists an integer $k$ such that $1\leq k<x$ and such that all the prime factors $p$ of $k^2x^2+1$ satisfy $p<x$.
I was able to verify this by computer for $x\leq10^6$.
This result would make it possible to solve the functional equation $f(n^2+1)-f(n^2)=1$ mentioned in this post.