We study whether sufficiently large integers can be written in the form cp+ Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2p−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rela...

In this paper, we verify the diameter of zero divisor graphs with respect to direct product. Keywords—Atomic lattice, complement of graph, diameter, direct product of lattices, 0-distributive lattice, girth, product of graphs, prime ideal, zero divisor graph.

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

By combining and improving recent techniques results, we provide explicit estimates for the error terms |π(x)−li(x)|, |θ(x)−x| |ψ(x)−x| appearing in prime number theorem. For example, show all x≥2 that |ψ(x)−x|≤9.39x(log⁡x)1.515exp⁡(−0.8274log⁡x). Our rely heavily on zero-free regions zero-density Riemann zeta-function, improve existing bounds prime-counting functions large values of x.

Let [Formula: see text] be a prime and let the complex field. We explicitly classify finite solvable irreducible monomial subgroups of up to conjugacy. That is, we give complete irredundant list text]-conjugacy class representatives as generating sets matrices. Copious structural information about non-solvable is also proved, enabling classification all such groups bar one family. explain obsta...

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by an Euler products and are analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in ...

In this paper we present a kind of group suitable for cryptographic applications: the trace zero subvariety. We describe in detail the case of trace zero varieties constructed from genus 2 curves over prime fields. The curve is considered over an extension field of degree 3 and one performs Weil descent from its Jacobian to the prime field leading to a variety of dimension 6. The trace zero var...

It was known that if one-way functions exist, then there are zero-knowledge proofs for every language in PSPACE . We prove that unless very weak one-way functions exist, ZeroKnowledge proofs can be given only for languages in BPP. For average-case de nitions of BPP we prove an analogous result under the assumption that uniform one-way functions do not exist. Thus, very loosely speaking, zero{kn...

It is shown that every commutative arithmetic ring R has λ-dimension ≤ 3. An example of a commutative Kaplansky ring with λ-dimension 3 is given. Moreover, if R satisfies one of the following conditions, semi-local, semi-prime, self f p-injective, zero-Krull dimensional, CF or FSI then λ-dim(R) ≤ 2. It is also shown that every zero-Krull dimensional commu-tative arithmetic ring is a Kaplansky r...