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Evaluation of a determinant involving Legendre symbols
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Evaluation of a determinant involving Legendre symbols
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Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely.
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Cited by 1 Pith paper
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A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei
Proves D_a(0)=0 iff p≡3 mod 4 and χ(a n!)=1, gives Pfaffian-square factorizations of the determinants for p≡3 mod 4, and settles Sun's conjecture when a=n!.
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