In this paper we study elementary approaches to classical theorems on representations of primes of the form ax2 + by2, in particular the two squares theorem. While most approaches make use of quadratic residues, we study a route initiated by Liouville, and simplified by Heath-Brown and Zagier.

We point out that off-shell four-dimensional spacetime-supersymmetry implies strange hermiticity properties for the N=1 RNS superstring. However, these hermiticity properties become natural when the N=1 superstring is embedded into an N=2 superstring.

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...

We construct a class of codes of length such that the minimum distance outside of a certain subcode is, up to a constant factor, bounded below by the square root of , a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizat...

131 W. E. Deskins, Abstract Algebra. London: Collier-Macmillan, 1964. [4] I. Flores, The Logic of Computer Arithmetic. Englewood Cliffs, N.J.: Prentice-Hall, 1963. 151 H. L. Garner, "The residue number system," IRE Trans. Electron. Comput., vol. EC-8, pp. 140-147, June 1959. 161 N. S. Szabo and R. 1. Tanaka, Residue Arithmetic and its Applications to Computer Technology. New York: McGraw-Hill, ...

A new method for speech parametric coding at medium outpul rate is presented. The availability of new el~ctronic components changes the hypotheses assumed in previous studies. A parametric coding scheme with medium outpul rate now seems appropriate for rule-based synthesis. The weak spot in most parametric synthesizers lies in the poor model used to represent the excitation signal. Our approach...

In this paper, we define a rough cosine similarity measure between two rough neutrosophic sets. The notions of rough neutrosophic sets (RNS) will be used as vector representations in 3D-vector space. The rating of all elements in RNS is expressed with the upper and lower approximation operator and the pair of neutrosophic sets which are characterized by truth-membership degree, indeterminacy-me...

In this paper, architecture of residue number system used in FIR filters, is presented. For many years residue number coding has been recognized as a system which provides capability for implementation of a high speed addition and multiplication. These advantages of residue number system coding for the high speed FIR filters design results from the fact that an digital FIR filter requires only ...