The Pell sequence and cyclotomic matrices involving squares over finite fields

TL;DR

This paper investigates cyclotomic-type matrices over finite fields built from sums of nonzero squares, analyzing when the matrices $B_q(m)=\left[(s_i+s_j)^m\right]$ are singular. By combining properties of the Pell sequence with $p$-adic tools, it obtains explicit determinant formulas and singularity criteria for $m= n-2, n-1, n$ with $n=(q-1)/2$, distinguishing the cases $f\ge2$ and $f=1$ (where $q=p$). The results tie determinant behavior to Pell sequence terms, e.g., $Q_p$ and $P_p$, yielding congruence-based conditions such as $Q_p\equiv 2\pmod{p^2}$ and relations involving $2P_p$ modulo $p^2$, and prompting a conjecture about the primes $p=13$ and $31$. Additionally, the paper develops a variant of Carlitz-type determinant results for Jacobi/Gauss-sum matrices using almost circulant matrices and the Gross–Koblitz formula.

Abstract

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.1
• Remark 1.1
• Conjecture 1.1
• Theorem 1.2
• Corollary 1.2
• Remark 1.2
• Theorem 1.3
• Corollary 1.3
• Lemma 2.1