On Some Determinants with Legendre Symbol Entries
نویسندگان
چکیده
Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 210093, People’s Republic of China [email protected] http://math.nju.edu.cn/∼zwsun Abstract. In this paper we mainly focus on some determinants with Legendre symbol entries. For an odd prime p and an integer d, let S(d, p) denote the determinant of the (p − 1)/2 × (p − 1)/2 matrix whose (i, j)-entry (1 6 i, j 6 (p− 1)/2) is the Legendre symbol ( i +dj p ). We investigate properties of S(d, p) as well as some other determinants involving Legendre symbols. In Section 3 we pose 17 open conjectures on determinants one of which states that ( −S(d,p) p ) = 1 if ( d p ) = 1, and S(d, p) = 0 if ( d p ) = −1. This material might interest some readers and stimulate further research.
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