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.

Parametrized units in algebraic number fields

1988 ◽  
Vol 103 (1) ◽  
pp. 15-25
Author(s):  
Loren D. Olson
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Algebraic Function ◽  
Number Fields ◽  
Algebraic Number Theory ◽  
Quadratic Fields ◽  
Integer Variable ◽  
Algebraic Number Fields ◽  
Algebraic Function Fields ◽  
Cubic Fields

One of the fundamental problems in algebraic number theory is the construction of units in algebraic number fields. Various authors have considered number fields which are parametrized by an integer variable. They have described units in these fields by polynomial expressions in the variable e.g. the fields ℚ(√[N2 + 1]), Nεℤ, with the units εN = N + √[N2 + l]. We begin this article by formulating a general principle for obtaining units in algebraic function fields and candidates for units in parametrized families of algebraic number fields. We show that many of the cases considered previously in the literature by such authors as Bernstein [2], Neubrand [8], and Stender [ll] fall in under this principle. Often the results may be obtained much more easily than before. We then examine the connection between parametrized cubic fields and elliptic curves. In §4 we consider parametrized quadratic fields, a situation previously studied by Neubrand [8]. We conclude in §5 by examining the effect of parametrizing the torsion structure on an elliptic curve at the same time.

scholarly journals On the System of Diophantine Equationsx2-6y2=-5andx=az2-b

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Silan Zhang ◽  
Jianhua Chen ◽  
Hao Hu
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Class Number ◽  
Diophantine Equations ◽  
Algebraic Number Theory ◽  
The Family ◽  
Integral Solutions

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions(x,y,z)of the system of Diophantine equationsx2-6y2=-5andx=2z2-1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equationsx2-6y2=-5andx=az2-bfor each pair of integral parametersa,b. The proof utilizes algebraic number theory andp-adic analysis which successfully avoid discussing the class number and factoring the ideals.

scholarly journals Class Number and Ramification in Number Fields

1963 ◽  
Vol 23 ◽  
pp. 97-101 ◽  
Author(s):  
Armand Brumer ◽  
Michael Rosen
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Algebraic Number ◽  
Class Number ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Algebraic Number Theory ◽  
Outstanding Problem ◽  
Principal Ideals

In the ring Ok of algebraic integers of a number field K the group Ik of ideals of Ok modulo the subgroup Pk of principal ideals is a finite abelian group of order hk, the class number of K. The determination of this number is an outstanding problem of algebraic number theory.

scholarly journals Nonexistence of Almost Moore Digraphs of Diameter Three

10.37236/811 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
J. Conde ◽  
J. Gimbert ◽  
J. Gonzàlez ◽  
J. M. Miret ◽  
R. Moreno
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Algebraic Number ◽  
Directed Graphs ◽  
Algebraic Number Theory ◽  
Cycle Structure ◽  
Spectral Techniques ◽  
Binary Matrices ◽  
Algebraic Multiplicities ◽  
Big Pieces

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

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