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

Srikanth Cherukupally

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.

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

Abstract

For number , let . We show that the size of is connected to a property concerning integer evaluations of Fibonacci-like polynomials. In the process, we prove that , and establish the average value of to be a little above , asymptotically. But the empirical data up to indicate that , proving which is left as an open issue.
Paper Structure (4 sections, 5 theorems, 7 equations)

This paper contains 4 sections, 5 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Fibonacci-like polynomials
  3. Chain of numbers
  4. Average value of $|\mathcal{A}(n)|$

Key Result

Lemma 2.1

For any $i\geq 1$, $k\geq 2$, $G_{i-1}(k) \in \mathcal{A}(G_i(k))$.

Theorems & Definitions (7)

  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Conjecture 3.4
  • Lemma 3.5
  • Lemma 3.6