Related papers: Chebotarev Sets
We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.
We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1].…
We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.
An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…
Given a number field $K$ that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers $\mathfrak{O}_K$ of $K$ by describing certain Frobenius semigroups,…
In this paper, we study the structure of finite groups with a large number of conjugacy classes of $p$-elements for some prime $p$. As consequences, we obtain some new criteria for the existence of normal $p$-complements in finite groups.
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.
We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and…
In this paper, we establish some criteria to detect the presence of the maximal ideal $(x_1, \ldots, x_n)$ in the set of associated primes of powers of monomial ideals in the polynomial ring $K[x_1, \ldots, x_n]$. Furthermore, for each of…
For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…
The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of…
The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…
Prime counting functions are believed to exhibit, in various contexts, discrepancies beyond what famous equidistribution results predict; this phenomenon is known as Chebyshev's bias. Rubinstein and Sarnak have developed a framework which…
For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all…
The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete…
This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and…
We characterize numerical semigroups for which the poset of its ideal class monoid is a lattice, and study the irreducible elements of such a lattice with respect to union, intersection, infimum and supremum.
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…