Cubic residues and binary quadratic forms
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- 作者：Zhi-Hong Sun
- 关键词：Cubic residue ; Binary quadratic form ; Cubic Jacobi symbol ; Cubic congruence
- 刊名：Journal of Number Theory
- 出版年：2007
- 出版时间：May 2007
- 年：2007
- 卷：124
- 期：1
- 页码：62-104
- 全文大小：363 K
文摘
Let p>3 be a prime, , gcd(u,v)=1, pu2−dv2 and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where a184ce9b9bb62801425a47d1"" title=""Click to view the MathML source"">{Un} is the Lucas sequence defined by 26a1e2a6f512e43"" title=""Click to view the MathML source"">U0=0, U1=1 and Un+1=PUn−QUn−1 (n1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.