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Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let Λ = b 2 log α 2 - b 1 log α 1 - b 3 log α 3 0 with b 1 , b 2 , b 3 positive integers and α 1 , α 2 , α 3 positive multiplicatively independent rational numbers greater than 1 . Let α j 1 = α j 1 / α j 2 with α j 1 , α j 2 coprime positive integers ( j = 1 , 2 , 3 ) . Let α j max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B max { 10 , log b ' } . We prove that either { b 1 , b 2 , b 3 } is c 4 , B -linearly dependent over (with respect to a 1 , a 2 , a 3 ) or Λ > exp - C B 2 j = 1 3 log a j , where c 4 and C = c 1 c 2 log ρ + δ are given in the tables...

Linear forms in two logarithms and interpolation determinants

Michel Laurent (1994)

Acta Arithmetica

1. Introduction. Our aim is to test numerically the new method of interpolation determinants (cf. [2], [6]) in the context of linear forms in two logarithms. In the recent years, M. Mignotte and M. Waldschmidt have used Schneider's construction in a series of papers [3]-[5] to get lower bounds for such a linear form with rational integer coefficients. They got relatively precise results with a numerical constant around a few hundreds. Here we take up Schneider's method again in the framework...

Lucas balancing numbers

Kálmán Liptai (2006)

Acta Mathematica Universitatis Ostraviensis

A positive n is called a balancing number if 1 + 2 + + ( n - 1 ) = ( n + 1 ) + ( n + 2 ) + + ( n + r ) . We prove that there is no balancing number which is a term of the Lucas sequence.

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

We show that the only Lucas numbers which are factoriangular are 1 and 2 .

Lucas sequences and repdigits

Hayder Raheem Hashim, Szabolcs Tengely (2022)

Mathematica Bohemica

Let ( G n ) n 1 be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are { U n } and { V n } , respectively. We show that the Diophantine equation G n = B · ( g l m - 1 ) / ( g l - 1 ) has only finitely many solutions in n , m + , where g 2 , l is even and 1 B g l - 1 . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on...

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