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.

On Some Exponential Equations of S. S. Pillai

2001 ◽  
Vol 53 (5) ◽  
pp. 897-922 ◽  
Author(s):  
Michael A. Bennett
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Prior Work ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Sharp Result ◽  
Exponential Equations ◽  
Positive Integers

AbstractIn this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ax – by = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

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.

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.

Solving the FCSR synthesis problem for multi-sequences by lattice basis reduction

2017 ◽  
Vol 86 (5) ◽  
pp. 1023-1038 ◽  
Author(s):  
Weihua Liu ◽  
Andrew Klapper ◽  
Zhixiong Chen
Keyword(s):  
Synthesis Problem ◽  
Lattice Basis Reduction ◽  
Basis Reduction

scholarly journals On explicit estimates for linear forms in the values of a class of E-functions

1982 ◽  
Vol 25 (1) ◽  
pp. 37-69 ◽  
Author(s):  
Xu Guangshan ◽  
Wang Lianxiang
Keyword(s):  
Differential Equations ◽  
Lower Bounds ◽  
Linear Differential Equations ◽  
Linear Forms ◽  
Linear Differential

We apply methods of Mahler to obtain explicit lower bounds for certain combinations of E-functions satisfying systems of linear differential equations as studied by Makarov. Our results sharpen and generalise earlier results of Mahler, Shidlovskii, and Väänänen.

Sign in / Sign up

Export Citation Format

Share Document