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

Abstract

For number $n>1$, let $\mathcal{A}(n)=\{1\le a<n:n|a^2-1,a|n^2-1\}$. 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 $|\mathcal{A}(n)|\le 3$, proving which is left as an open issue.