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Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure of Gauss's reduction theory into that of Zagier's. From this perspective, Zagier-reduction becomes a refinement of Gauss-reduction. [ABSTRACT FROM AUTHOR]

In this article we estimate the number of integers up to X which can be properly represented by a positive-definite, binary, integral quadratic form of small discriminant. This estimate follows from understanding the vector of signs that arises from computing the Legendre symbol of small integers n at multiple primes. [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper we show that p can be expressed as p = a 2 ± a b − b 2 with integers a and b if and only if p is congruent to 0, 1 or −1 ( mod 5 ) and p can be expressed as p = a 2 ± a b + b 2 with integers a and b if and only if p is congruent to 0, 1 ( mod 3 ) . [ABSTRACT FROM AUTHOR]

Abstract: Let be a prime, and let m be an integer with . In the paper we solve some conjectures of Z.W. Sun concerning , and . In particular, we show that for . Let be the Legendre polynomials. In the paper we also show that , where t is a rational p-adic integer, is the greatest integer not exceeding x and is the Legendre symbol. As consequences we determine in the cases and confirm many conjectures of Z.W. Sun. [Copyright &y& Elsevier]

Abstract: Let be a prime, and let m be an integer with . In this paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuringʼs theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning . [Copyright &y& Elsevier]

Abstract: Let a and b be positive integers and let p be an odd prime such that for some integers x and y. Let be given by . In this paper, using Jacobiʼs identity , we construct in terms of . For example, if and , then . We also give formulas for , and . [Copyright &y& Elsevier]

Abstract: Let be a prime, , , 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 to be a cubic residue of p, where is the fundamental unit of the quadratic field . We also give a general criterion for , where is the Lucas sequence defined by , and (). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. [Copyright &y& Elsevier]

Abstract: Let be a prime, and . In this paper we obtain a general criterion for to be a quartic residue in terms of appropriate binary quadratic forms. Let be a squarefree integer such that , where is the Legendre symbol, and let be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes so that is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where is an odd prime, and . As an application we also obtain a general criterion for , where is the Lucas sequence defined by and . [Copyright &y& Elsevier]