The Pell sequence and cyclotomic matrices involving squares over finite fields
Volume 224 / 2026
Acta Arithmetica 224 (2026), 1-25
MSC: Primary 11L05; Secondary 15A15, 11R18, 12E20
DOI: 10.4064/aa250311-18-10
Published online: 27 May 2026
Abstract
By using 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 $1=s_1,s_2,\ldots ,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb {F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge 7$. We prove that the matrix $$ B_q((q-3)/2)=[(s_i+s_j)^{(q-3)/2}]_{2\le i,j\le (q-1)/2} $$ is singular whenever $f\ge 2$. Also, for $q=p$, we show that $$ \det B_p((p-3)/2)=0\iff Q_p\equiv 2\pmod{p^2\mathbb {Z}}, $$ 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 $Q_{i+1}=2Q_i+Q_{i-1}$.
