Evaluation of a determinant involving Legendre symbols

Abstract

Let $p>3$ be a prime, and let $(\frac{\cdot }{p})$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}{]}_{1⩽i,j⩽(p-1)/2}$, where $$a_{ij}=\{\begin{array}{llc}(\frac{j}{p})& \text{if}i=1,\$\frac{i+j}{p})& \text{if}i>1.\end{array}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3(\mathrm{mod}4)$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely.

14 pages