Related papers: An estimate for Frobenius' diophantine problem in …

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer not belonging to the set $\big\{ \sum_{i=1}^na_ix_i\ |x_i\in \mathbb{N}\big\}$. The general Frobenius…

Let $A=(a_1, a_2, \ldots, a_n)$ be a sequence of relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus…

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

We refine a result of W.P. Li and Wang on the values of the form $ \lambda_1p_1 + \lambda_2p_2^{2} + \lambda_3p_3^{2} + \mu_1 2^{m_1} +...+ \mu_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers,…

Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of…

Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We…

In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of…

For $ k \geq 2 $, let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) = 1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s) =…

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…

Let $\lambda_i, \mu_j$ be non-zero real numbers not all of the same sign and let $a_i, b_k$ be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape \begin{equation*} \begin{cases} \left|…

This paper addresses a problem recently raised by Laurent and Nogueira about inhomogeneous Diophantine approximation with coprime integers. As a corollary of our main theorem we obtain an improvement of the best known exponent of…

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in…

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a…

We consider the space $[0,n]^3$, imagined as a three dimensional, axis-aligned grid world partitioned into $n^3$ $1\times 1 \times 1$ unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it,…

We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.

Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients,…

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural…

We consider the Diophantine problem of Frobenius for semigroup ${\sf S}({\bf d}^3)$ where ${\bf d}^3$ denotes the tuple $(d_1,d_2,d_3)$, $\gcd(d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic…

We give an upper bound for the minimum $s$ with the property that every sufficiently large integer can be represented as the sum of $s$ positive $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq…