An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieris theorem on primes in arithmetic progressions.     

On arithmetic progressions in recurrences – A new characterization of the Fibonacci sequence

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given.     

On arithmetic progressions on Pellian equations

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x ² ? dy ² = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.     

Discrepancy One among Homogeneous Arithmetic Progressions

We investigate a restriction of Paul Erd?s’ well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the discrepancy with respect to this set is exactly 1. We answer this question when S has size four or less, and prove that the problem for general S is NP-hard, even for discrepancy 1.     

On the length of arithmetic progressions in linear combinations of S-units

Recent finiteness results concerning the lengths of arithmetic progressions in linear combinations of elements from finitely generated multiplicative groups have found applications to a variety of problems in number theory. In the present paper, we significantly refine the existing arguments and give an explicit upper bound on the length of such progressions.     

Generalized van der Waerden numbers

Certain generalizations of arithmetic progressions are used to define numbers analogous to the van der Waerden numbers. Several exact values of the new numbers are given, and upper bounds for these numbers are obtained. In addition, a comparison is made between the number of different arithmetic progressions and the number of different generalized arithmetic progressions.     

Connections between connected topological spaces on the set of positive integers

In this paper we introduce a connected topology T on the set ? of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ? which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (?, T) and (?, T′).     

Powerful arithmetic progressions

We give a complete characterization of so-called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length.     

Additive functions on arithmetic progressions with large moduli

In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott.     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

On Irreducibles in the Polynomial Semigroup

We consider the asymptotic behavior of the number of irreducible polynomials over a finite field in arithmetic progressions.     

The closures of arithmetic progressions in the common division topology on the set of positive integers

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

On the density of integers of the form 2 k + p in arithmetic progressions

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.     

Large sieve inequality with sparse sets of moduli applied to Goldbach conjecture

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.     

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.     

On mixed joint discrete universality for a class of zeta-functions. II

We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used.

     

A variational Barban–Davenport–Halberstam Theorem

We prove variational forms of the Barban–Davenport–Halberstam Theorem and the large sieve inequality. We apply our result to prove an estimate for the sum of the squares of prime differences, averaged over arithmetic progressions.     

Coloured solutions of equations in finite groups

In this article, we consider the relations between colourings and some equations in finite groups. We will express relations linking the numbers of the differently coloured solutions of an equation that depend only on the cardinality of the colouring and not on the distribution of the colours. This gives a link between Ramsey theory that investigates the existence of monochromatic solutions and what is now called anti-Ramsey theory that investigates the existence of rainbow solutions. Both theories are in expansion. We will apply these results to the counting of rainbow 3-term arithmetic progressions in any abelian group with equinumerous three-colouring and to the counting of points on a conic defined on a finite field. We will end by discussing the generalised case of a system of equations.