On certain determinants and the square root of some determinants involving Legendre Symbols

2024/07/05 by Ren, Chen-kai, Luo, Xin-qi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.04556

Abstract

Let p>3 be a prime and ((.)/(p)) be the Legendre symbol. For any integer d with p\nmid d and any positive integer m, Sun introduced the determinants T_m(d,p)=det[(i²+dj²)^m((i²+dj²)/(p))]_{1\leqslant i,j \leqslant (p-1)/2}, and D_p^(m)= det[(i²-j²)^m((i²-j²)/(p))]_{1≤ i,j≤ (p-1)/2} . In this paper, we obtain some properties of T_m (d,p) and √D_p^(m) for some m. We also confirm some related conjectures posed by Zhi-Wei Sun.

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• Determinants concerning Legendre symbols
• On some determinants arising from quadratic residues
• Problems and results on determinants involving Legendre symbols
• Proof of three conjectures on determinants related to quadratic residues
• Legendre symbols related to D_p(b,1)