Related papers: On a Fundamental Task of Diophantine Geometric Fig…
The goal of the work is to take on and study one of the fundamental tasks studying Bidiophantine polygons (let us call a polygon Diophantine, if the distance between each two vertex of those is expressed by a natural number and we say that…
In this article one of the fundamental problems of Diophantine (integer) planar geometric figures, the Task max n(k), is considered.
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
Over the last two hundred years different transformation formulas for Gauss' hypergeometric function ${}_2F_1$ were discovered. The goal of the present article is to study their arithmetic analogue for the underlying hypergeometric motive.…
We give a brief exposition on the uses of non-commutative fundamental groups for the study of Diophantine problems via a non-abelian Albanese map.
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…
The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
In this paper Gauss integers are used for to develop some geometry of full Erdoes graphs which vertics are points with integer coordinates. These graphs are called Diophantine figures.
We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.
In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…
The author surveys the problem of piecing together integral or rational solutions to Diophantine equations (global structure) from solutions modulo congruences and real solutions (local structure).
A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1x_1+\cdots +f_nx_n \mod b \le g_1x_1+\cdots +g_nx_n$ where $g_1,\dots,g_n,$ $f_1,\ldots ,f_n\in \mathbb{Z}$ and…
Let $\Gamma$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of…
This paper explores the integration of Diophantine equations into neural network (NN) architectures to improve model interpretability, stability, and efficiency. By encoding and decoding neural network parameters as integer solutions to…
This survey paper is not a complete reference guide to number-theoretical applications of ergodic theory. Instead, it considers an approach to a class of problems involving Diophantine properties of $n$-tuples of real numbers, namely,…
This paper is the ninth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic…
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\'an Bertalan and we also provide some generalization.
The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for…