Congruences concerning Legendre polynomials II

JOURNAL OF NUMBER THEORY

卷:133

Article

Sun, Zhi-Hong

关键词: Legendre polynomial;    Congruence;    Character sum;    Binary quadratic form;    Elliptic curve;   

DOI  :  10.1016/j.jnt.2012.11.004

来源: Elsevier

【 摘 要 】

Let p > 3 be a prime, and let m be an integer with p inverted iota m. In the paper we solve some conjectures of Z.W. Sun concerning Sigma(p-1)(k=0) ((2k)(k))(3) /m(k) p(2)), Sigma(p-1)(k=0) ((2k)(k))((4k)(2k))/m(k) (mod p) and Sigma(p-1)(k=0) ((2k)(k))(2) ((4k)(2k))/m(k) (mod p(2)). In particular, we show that Sigma(p-1/2)(k=0) ((2k)(k))(3) equivalent to 0 (mod p(2)) for p equivalent to 3, 5, 6 (mod 7). Let {P-n(x)} be the Legendre polynomials. In the paper we also show that P-[p/4] (t) equivalent to -(6/p) Sigma(p-1)(x=0)(x(3)-3/2(3t+5)x+9t+7)/p (mod p), where t is a rational p-adic integer, [x] is the greatest integer not exceeding x and (a/p) is the Legendre symbol. As consequences we determine P-[p/4](t) (mod p) in the cases t = -5/3, -7/9, -65/63 and confirm many conjectures of Z.W. Sun. (C) 2013 Elsevier Inc. All rights reserved.

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