相关论文: Rational Quartic Reciprocity
We continue investigating rational quartic reciprocity laws and, at the suggestion of the editor of AA, provide details of a proof of a remark in the first article with this title.
We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.
The shortest known proof of the law of quadratic reciprocity (without supplements) is presented.
Using the quadratic reciprocity law as the motivating example, we convey an understanding of classical reciprocity laws.
A proof of the Quadratic Reciprocity Law is presented using a Lemma of Gauss, the theory of finite fields and the Frobenius automorfism.
In this paper, we establish some reciprocity formulas for certain generalized Hardy-Berndt sums by using the Fourier series technique and some properties of the periodic zeta function and the Lerch zeta function. It turns out that one of…
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
We present a creative reimagining of Zolotarev's classical proof of the Law of Quadratic Reciprocity.
In this article we define a quadratic symbol for a finite group and prove a law of reciprocity for its value.
We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the…
We give a reciprocity formula for a two-variable sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is approximable from below and show that the reciprocity formula…
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…
The paper contained a preliminary version of a general theory of reciprocity laws on vector spaces.
In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's…
We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.
Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.
In this paper, for coprime numbers p and q we consider the well known Dedekind sums S(p,q) First, we give an improvement of the proof given by H. Rademacher and A. Whiteman, and we construct a new arithmetical proof for the reciprocity law
In this paper, we study the generalized Dedekind-Rademacher sums considered by Hall, Wilson and Zagier. We establish a formula for the products of two Bernoulli functions. The proof relies on Parseval's formula, Hurwitz's formula, and…
Riemann's non-differentiable function and Gauss's quadratic reciprocity law have attracted the attention of many researchers. In \cite{RM} Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta-transformation…
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…