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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 ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( p). [ABSTRACT FROM AUTHOR]

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( p). [ABSTRACT FROM AUTHOR]

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]