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Determinants concerning Legendre symbols

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 نشر من قبل Hai-Liang Wu
 تاريخ النشر 2020
  مجال البحث
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 تأليف Hai-Liang Wu




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The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $aequiv1pmod4$. Then for the determinant $$S(1,p):={rm det}bigg[left(frac{i^2+j^2}{p}right)bigg]_{1le i,jle frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.



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