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For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of…

We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional…

In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction…

We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in \S 4 and then in \S 5 we give a new proof for…

In this paper we show that Diophantine problem for quadratic equations in Baumslag-Solitar groups $BS(1,k)$ and in wreath products $A \wr \mathbb{Z}$, where $A$ is a finitely generated abelian group and $\mathbb{Z}$ is an infinite cyclic…

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic…

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group…

The behavior of the images of a fixed element of order p in irreducible representations of a classical algebraic group in odd characteristic p with highest weights large enough with respect to p and this element is investigated. Lower…

Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy…

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…

Let l be an odd prime. For primes, p \equiv 1 (mod l), Gauss (l = 3) and Dickson (l = 5) considered the Diophantine systems in terms of which cyclotomic numbers of order 3 and 5 were obtained. The aim of this paper is to show how to obtain…

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II…

Julius Whiston and Jan Saxl showed that the size of an irredundant generating set of the group G=PSL(2,p) is at most four and computed the size m(G) of a maximal set for many primes. We will extend this result to a larger class of primes,…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

We investigate the distribution of $p\theta$ modulo 1, where $\theta$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a…

Two Diophantine equation generator function for integer residuals produced by integer division over closed intervals are presented. One each for the closed intervals [1,Floor(n^0.5)] and [Ceiling(n^0.5),n], respectively.

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic…