Quartic, octic residues and Lucas sequences

JOURNAL OF NUMBER THEORY

卷:129

Article

Sun, Zhi-Hong

关键词: Lucas sequence;    Congruence;    Quartic Jacobi symbol;   

DOI  :  10.1016/j.jnt.2008.09.019

来源: Elsevier

【 摘 要 】

Let p equivalent to 1 (mod 4) be a prime and a, b is an element of Z with a(2) + b(2) not equal p. Suppose p = x(2) + (a(2) + b(2))y(2) for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine (b+root a(2)+b(2))/2 (p-1/4) (mod P). As applications we obtain the congruences for U (p-1/4) modulo p and the criteria for p vertical bar U (p-1/8) (if p equivalent to 1 (mod 8)), where [U-n] is the Lucas sequence given by U-0=0, U-1 = 1 and Un+1 =bU(n) + k(2)U(n-1) (n >= 1). We also pose many conjectures concerning U (p-1/4), m (p-1/8) or m (p-5/8) (mod P). (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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