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

TL;DR

This paper investigates the size of a specific set related to divisibility conditions involving a number n, revealing its connection to Fibonacci-like polynomials, and establishes bounds and average behavior with empirical evidence up to 10^7.

Contribution

It introduces a novel connection between the set size and Fibonacci-like polynomial evaluations, providing bounds and asymptotic average estimates, and highlights an open problem regarding the maximum size.

Findings

|A(n)| < log_2 n for all n > 1

Average |A(n)| is slightly above 2 asymptotically

Empirical data suggests |A(n)| ≤ 3 for n < 10^7

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.