Two topics in number theory. Diophantine equations with infinite solutions and a theorem on factorization of integers

2016 ◽  
Vol 11 ◽  
pp. 463-470
Author(s):  
Rafael Jakimczuk
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Factorization Of Integers

scholarly journals Solving Diophantine Equations by Factoring in Number Fields

2021 ◽  
Vol 2 (1) ◽  
pp. 29-34
Author(s):  
Zdeněk Pezlar
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Algebraic Number Theory ◽  
Quadratic Residues ◽  
Worked Problems

In this text we provide an introduction to algebraic number theory and show its applications in solving certain difficult diophantine equations. We begin with a quick summary of the theory of quadratic residues, before diving into a select few areas of algebraic number theory. Our article is accompanied by a couple of worked problems and exercises for the reader to tackle on their own.

scholarly journals Lattices and the Geometry of Numbers

2021 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Functional Analysis ◽  
Number Theory ◽  
Diophantine Equations ◽  
Algebraic Numbers ◽  
Geometry Of Numbers ◽  
Geometrical Properties ◽  
The Subject ◽  
Broad Overview

In this paper, we discuss the properties of lattices and their application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has an application in various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis, etc. This paper gives an elementary introduction to the field of the geometry of numbers. In this paper, we shall first give a broad overview of the concept of lattice and then discuss the geometrical properties it has and its applications.

scholarly journals Constructing Fifteen Infinite Classes of Nonregular Bipartite Integral Graphs

10.37236/732 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ligong Wang ◽  
Cornelis Hoede
Keyword(s):  
Number Theory ◽  
Adjacency Matrix ◽  
Diophantine Equations ◽  
Matrix Theory ◽  
Discrete Math ◽  
Computer Search ◽  
Characteristic Polynomials ◽  
Integral Property ◽  
Open Problems ◽  
Integral Graphs

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs $S_1(t)=K_{1,t}$, $S_2(n,t)$, $S_3(m,n,t)$, $S_4(m,n,p,q)$, $S_5(m,n)$, $S_6(m,n,t)$, $S_8(m,n)$, $S_9(m,n,p,q)$, $S_{10}(n)$, $S_{13}(m,n)$, $S_{17}(m, n, p, q)$, $S_{18}(n,p,q,t)$, $S_{19}(m,n,p,t)$, $S_{20}(n,p,q)$ and $S_{21}(m,t)$ are defined. We construct the fifteen classes of larger graphs from the known 15 smaller integral graphs $S_1-S_6$, $S_8-S_{10}$, $S_{13}$, $S_{17}-S_{21}$ (see also Figures 4 and 5, Balińska and Simić, Discrete Math. 236(2001) 13-24). These classes consist of nonregular and bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. We obtain their integral property by using number theory and computer search. All these classes are infinite. They are different from those in the literature. It is proved that the problem of finding such integral graphs is equivalent to solving Diophantine equations. We believe that it is useful for constructing other integral graphs. The discovery of these integral graphs is a new contribution to the search of integral graphs. Finally, we propose several open problems for further study.

scholarly journals Lattices and the Geometry of Numbers

2021 ◽  
pp. 77-92
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Functional Analysis ◽  
Number Theory ◽  
Diophantine Equations ◽  
Algebraic Numbers ◽  
Geometry Of Numbers ◽  
Geometrical Properties ◽  
The Subject ◽  
Broad Overview

In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory.  This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper gives an elementary introduction to the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications.

scholarly journals EQUAÇÕES DIOFANTINAS: UM PROJETO PARA A SALA DE AULA E O USO DO GEOGEBRA

2015 ◽  
Vol 37 ◽  
pp. 532
Author(s):  
Alexandre Hungaro Vansan
Keyword(s):  
High School ◽  
Number Theory ◽  
Diophantine Equations ◽  
Lesson Plan ◽  
Solution Set ◽  
Algebraic Equations ◽  
Further Training ◽  
School Life ◽  
Work Plan ◽  
Linear Diophantine Equations

http://dx.doi.org/10.5902/2179460X14629The study of Number Theory here in this article aims to study some properties of integer multiples or divisors, emphasizing issues related to divisibility, which will be of great importance for the study of Diophantine equations, which in turn will provide for applications using Geogebra software. The Diophantine equations are algebraic equations that show the solution set of integers, which in this paper we will discuss the Linear Diophantine equations with two unknowns of the form 𝑎 x + 𝑏 y = 𝑐 with 𝑎, 𝑏, 𝑐 integers. In which they are applied as an alternative way for students to find solutions to problems he faced during his school life. This work is intended to further training of teachers who are teaching in the elementary and high school, where you can find suggestions for activities that you can apply in the classroom, or even include in your lesson plan Diophantine equations, since here he will find a suggestion of teaching work plan to include in their classes.

Some strongly undecidable natural arithmetical problems, with an application to intuitionistic theories

2003 ◽  
Vol 68 (1) ◽  
pp. 262-266
Author(s):  
Panu Raatikainen
Keyword(s):  
Number Theory ◽  
Essential Role ◽  
Diophantine Equations ◽  
Formal Learning ◽  
Trial And Error ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Formal Learning Theory ◽  
Martin Davis ◽  
Ordinary Number

Although Church and Turing presented their path-breaking undecidability results immediately after their explication of effective decidability in 1936, it has been generally felt that these results do not have any direct bearing on ordinary mathematics but only contribute to logic, metamathematics and the theory of computability. Therefore it was such a celebrated achievement when Yuri Matiyasevich in 1970 demonstrated that the problem of the solvability of Diophantine equations is undecidable. His work was building essentially on the earlier work by Julia Robinson, Martin Davis and Hilary Putnam (1961), who had showed that the problem of solvability of exponential Diophantine equations is undecidable. One should note, however, that although it was only Matiyasevich's result which finally solved Hilbert's tenth problem, already the earlier result had provided a perfectly natural problem of ordinary number theory which is undecidable.Nevertheless, both the set of Diophantine equations with solutions and the set of exponential Diophantine equations with solutions are still semi-decidable, that is, recursively enumerable (i.e., Σ10); if an equation in fact has a solution, this can be eventually verified. More generally, they are — as are their complements, the sets of equations with no solutions, which are Π10, — also Trial and Error decidable (Putnam [1965]), or decidable in the limit (Shoenfield [1959]), for every Δ20 set is (and conversely). This last-mentioned natural “liberalized” notion of decidability has begun more recently to play an essential role e.g., in so-called Formal Learning Theory (see e.g., Osherson, Stob, and Weinstein [1986], Kelly [1996]).

Representations of a rational number as a sum of odd powers

2016 ◽  
Vol 12 (04) ◽  
pp. 903-911
Author(s):  
M. A. Reynya
Keyword(s):  
Number Theory ◽  
Rational Number ◽  
Diophantine Equations ◽  
Symmetric Form ◽  
Parametric Solutions ◽  
Odd Degree

In this paper, we substantially generalize one of the results obtained in our earlier paper [M. A. Reynya, Symmetric homogeneous Diophantine equations of odd degree, Int. J. Number Theory 28 (2013) 867–879]. We present a solution to a problem of Waring type: if [Formula: see text] is a symmetric form of odd degree [Formula: see text] in [Formula: see text] variables, then for any [Formula: see text], [Formula: see text], the equation [Formula: see text] has rational parametric solutions, that depend on [Formula: see text] parameters.

Questions of decidability and undecidability in Number Theory

1994 ◽  
Vol 59 (2) ◽  
pp. 353-371 ◽  
Author(s):  
B. Mazur
Keyword(s):  
Number Theory ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Computable Function ◽  
Integral Solution ◽  
Survey Article ◽  
Image Position ◽  
Parameter Values ◽  
Computable Functions ◽  
Integral Solutions

Davis, Matijasevic, and Robinson, in their admirable survey article [D-M-R], interpret the negative solution of Hilbert's Tenth Problem as a resounding positive statement about the versatility of Diophantine equations (that any listable set can be coded as the set of parameter values for which a suitable polynomial possesses integral solutions).One can also view the Matijasevic result as implying that there are families of Diophantine equations parametrized by a variable t, which have integral solutions for some integral values t = a > 0, and yet there is no computable function of t which provides an upper bound for the smallest integral solution for these values a. The smallest integral solutions of the Diophantine equation for these values are, at least sporadically, too large to be bounded by any computable function. This is somewhat difficult to visualize, since there is quite an array of computable functions. But let us take an explicit example. Consider the functionMatijasevic's result guarantees the existence of parametrized families of Diophantine equations such that even this function fails to yield an upper bound for its smallest integral solutions (for all values of the parameter t for which there are integral solutions).Families of Diophantine equations in a parameter t, whose integral solutions for t = 1, 2, 3,… exhibit a certain arythmia in terms of their size, have fascinated mathematicians for centuries, and this phenomenon (the size of smallest integral solution varying wildly with the parameter-value) is surprising, even when the equations are perfectly “decidable”.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

The Postage–Stamp Problem, Number Theory, and the Programmable Calculator

1999 ◽  
Vol 92 (1) ◽  
pp. 20-22
Author(s):  
Harris S. Shultz
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Postage Stamp ◽  
Programmable Calculator ◽  
Problem Number

The “postage-stamp problem” is a classic question in the area of Diophantine equations. Students can explore this rich and accessible problem to deepen their understanding of linear equations.

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