LINEAR FORMS IN THE VALUES OF $ G$-FUNCTIONS, AND DIOPHANTINE EQUATIONS

1983 ◽  
Vol 45 (3) ◽  
pp. 379-396
Author(s):  
E M Matveev
Keyword(s):  
Diophantine Equations ◽  
Linear Forms

scholarly journals A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

2008 ◽  
Vol 60 (3) ◽  
pp. 491-519 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek
Keyword(s):  
Lower Bounds ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Arbitrary Degree ◽  
Famous Theorem ◽  
Linear Forms ◽  
Positive Integers ◽  
Level Lowering

AbstractWe solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation5uxn − 2r3s yn = ±1,in non-zero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

scholarly journals On a few Diophantine equations, in particular, Fermat's last theorem

2003 ◽  
Vol 2003 (71) ◽  
pp. 4473-4500
Author(s):  
C. Levesque
Keyword(s):  
Diophantine Equations ◽  
Number Fields ◽  
Linear Forms In Logarithms ◽  
Open Problems ◽  
Abc Conjecture ◽  
Linear Forms ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
The Subject ◽  
Integral Solutions

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solvedà laWiles. We will exhibit many families of Thue equations, for which Baker's linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, theABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

scholarly journals Linear forms in logarithms and exponential Diophantine equations

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Rob Tijdeman
Keyword(s):  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Exponential Diophantine Equations ◽  
International Audience

International audience This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

Reducible Diophantine Equations and Their Parametric Representations

1955 ◽  
Vol 7 ◽  
pp. 328-336
Author(s):  
E. Rosenthall
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Parametric Representation ◽  
Rational Integer ◽  
Homogeneous Polynomials ◽  
Linear Forms ◽  
Rational Field ◽  
Integer Solutions ◽  
Image Position ◽  
General Method

1. Reducible diophantine equations. The present paper will provide a general method for obtaining the complete parametric representation for the rational integer solutions of the multiplicative diophantine equation1.1 for some specified range of k, where the aijk,bijk are non-negative integers and the fki, hki are decomposable forms, that is to say they are integral irreducible homogeneous polynomials over the rational field R of degree k in k variables which can be written as the product of k linear forms.

The S-unit equation over function fields

1984 ◽  
Vol 95 (1) ◽  
pp. 3-4 ◽  
Author(s):  
Joseph H. Silverman
Keyword(s):  
Algebraic Geometry ◽  
Diophantine Equations ◽  
Function Fields ◽  
Number Fields ◽  
Upper Bounds ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Image Position ◽  
The Given ◽  
Integral Solutions

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equationin S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.

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