On the size of $\{a: 1\leq a<n, n|a^2-1, a|n^2-1\}$ for number $n$

Abstract

For number $n>1$, let $\mathcal{A}(n) = \{1\leq a<n: n|a^2-1, a|n^2-1 \}$. We show that the size of $\mathcal{A}(n)$ is connected to a property concerning integer evaluations of Fibonacci-like polynomials. In the process, we prove that $|\mathcal{A}(n)|< \log_2 n$, and establish the average value of $|\mathcal{A}(n)|$ to be a little above $2$, asymptotically. But the empirical data up to $n<10^7$ indicate that $|\mathcal{A}(n)|\leq 3$, proving which is left as an open issue.