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.

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.

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.

INTEGRAL POINTS ON CONGRUENT NUMBER CURVES

2013 ◽  
Vol 09 (06) ◽  
pp. 1619-1640 ◽  
Author(s):  
MICHAEL A. BENNETT
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Precise Description ◽  
Integral Points ◽  
Linear Forms ◽  
Integer Points ◽  
Congruent Number

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 - N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of (non-trivial) computation.

scholarly journals On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II

1984 ◽  
Vol Volume 7 ◽  
Author(s):  
Tarlok Nath Shorey
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience Using the theory of linear forms in logarithms we generalize an earlier result with R. Balasubramanian on the equation of the title.

scholarly journals Effective irrationality measures for quotients of logarithms of rational numbers

2015 ◽  
Vol Volume 38 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Rational Numbers ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience We establish uniform irrationality measures for the quotients of the logarithms of two rational numbers which are very close to 1. Our proof is based on a refinement in the theory of linear forms in logarithms which goes back to a paper of Shorey.

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