There is a rich literature on conditions guaranteeing that certain sums of residue classes cover certain other residue classes modulo some integer N . Often it is of particular importance for applications to know that the class 0 mod N can be represented as a sum of the studied residue classes, and the zero class is often the most difficult case. A well-known result along these lines is the fam...

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly dis...

This article presents a novel fault-tolerant technique which is based on the previously introduced Modulus Replication Residue Number System (MRRNS). Using the MRRNS mapping technique, we are able to compute using modular arithmetic over identical channels. We can make this system fault tolerant by increasing the number of channels, but by using some elementary polynomial properties we are able...

Using the Newton’s identities, we give the inductive formula for the generator polynomials of the p-adic quadratic residue codes.

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identiication we deduce systems of congruences which are analogues and generalizations of the Ku...