On some determinants arising from quadratic residues

2024/04/17 by Ren, Chen-Kai, Sun, Zhi-Wei
#11A15 #11C20 #15A15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.11547

Abstract

Let p>3 be a prime, and let d∈\mathbb Z with p\nmid d. For the determinants S_m(d,p)=det[(i²+dj²)^m]_{1\leqslant i,j \leqslant (p-1)/2} (\fracp-12\leqslant m\leqslant p-1), Sun recently determined S_m(d,p) modulo p when m∈\p-2,p-3\ and (\frac -dp)=-1. In this paper, we obtain S_p-2(d,p) modulo p in the remaining case (\frac-dp)=1, and determine the Legendre symbols (\fracS_p-3 (d,p)p) and (\fracS_p-4 (d,p)p) in some special cases.

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