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 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 Mordell’s equation: a classical approach

2015 ◽  
Vol 18 (1) ◽  
pp. 633-646 ◽  
Author(s):  
Michael A. Bennett ◽  
Amir Ghadermarzi
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Classical Approach ◽  
Linear Forms In Logarithms ◽  
Lattice Basis Reduction ◽  
Linear Forms ◽  
Basis Reduction

We solve the Diophantine equation$Y^{2}=X^{3}+k$for all nonzero integers$k$with$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

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.

Simultaneous approximation of numbers connected with the exponential function

1978 ◽  
Vol 25 (4) ◽  
pp. 466-478 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Lower Bounds ◽  
Exponential Function ◽  
Complex Number ◽  
Simultaneous Approximation ◽  
Complex Numbers ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms

AbstractWe give several results concerning the simultaneous approximation of certain complex numbers. For instance, we give lower bounds for |a–ξo |+ | ea – ξ1 |, where a is any non-zero complex number, and ξ are two algebraic numbers. We also improve the estimate of the so-called Franklin Schneider theorem concerning | b – ξ | + | a – ξ | + | ab – ξ. We deduce these results from an estimate for linear forms in logarithms.

Fibonacci numbers in generalized Pell sequences

2020 ◽  
Vol 70 (5) ◽  
pp. 1057-1068
Author(s):  
Jhon J. Bravo ◽  
Jose L. Herrera
Keyword(s):  
Lower Bounds ◽  
Continued Fractions ◽  
Fibonacci Numbers ◽  
Independent Interest ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Pell Numbers

AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

scholarly journals Quelques remarques sur des questions d'approximation diophantienne

1999 ◽  
Vol 59 (2) ◽  
pp. 323-334 ◽  
Author(s):  
Patrice Philippon
Keyword(s):  
Lower Bounds ◽  
Approximation Theory ◽  
Diophantine Approximation ◽  
Rational Numbers ◽  
Product Theorem ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Subspace Theorem

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

scholarly journals Tribonacci numbers that are concatenations of two repdigits

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Mahadi Ddamulira
Keyword(s):  
Lower Bounds ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Reduction Procedure ◽  
Linear Forms ◽  
Tribonacci Numbers

Abstract Let $$ (T_{n})_{n\ge 0} $$ ( T n ) n ≥ 0 be the sequence of Tribonacci numbers defined by $$ T_0=0 $$ T 0 = 0 , $$ T_1=T_2=1$$ T 1 = T 2 = 1 , and $$ T_{n+3}= T_{n+2}+T_{n+1} +T_n$$ T n + 3 = T n + 2 + T n + 1 + T n for all $$ n\ge 0 $$ n ≥ 0 . In this note, we use of lower bounds for linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction procedure to find all Tribonacci numbers that are concatenations of two repdigits.

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