# Linear Forms in Logarithms and Applications

@inproceedings{Bugeaud2018LinearFI,
title={Linear Forms in Logarithms and Applications},
author={Yann Bugeaud},
year={2018}
}
Repdigits as products of two Fibonacci or Lucas numbers
• Mathematics
• 2020
In this study, we show that if $$2\le m\le n$$ 2 ≤ m ≤ n and $$F_{m}F_{n}$$ F m F n represents a repdigit, then ( m ,  n ) belongs to the set \begin{aligned} \left\{ Mahler's work on Diophantine equations and subsequent developments • Mathematics • 2018 We discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, S-unit equations and S-integral points on elliptic curves, An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers. In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number. On the Location of Roots of the Characteristic Polynomial of (p, q)-Distance Fibonacci Sequences • Mathematics Bulletin of the Brazilian Mathematical Society, New Series • 2022 Extremely Fast Convergence Rates for Extremum Seeking Control with Polyak-Ruppert Averaging • Computer Science • 2022 It is shown in this paper that through design it is possible to obtain far faster convergence, of order O ( n − 4+ δ ), with δ > 0 arbitrary. Fractional parts of powers of real algebraic numbers • Y. Bugeaud • Mathematics Comptes Rendus. Mathématique • 2022 Let α be a real algebraic number greater than 1. We establish an effective lower bound for the distance between an integral power of α and its nearest integer. On the upper bound of the L_2-discrepancy of Halton’s sequence Let (H(n))n≥0 be a 2−dimensional Halton’s sequence. Let D2((H(n)) N−1 n=0 ) be the L2-discrepancy of (Hn) N−1 n=0 . It is known that lim supN→∞(logN) D2(H(n)) N−1 n=0 > 0. In this paper, we prove On a variant of Pillai's problem with transcendental numbers • Mathematics Acta Mathematica Hungarica • 2022 We study the asymptotic behaviour of the number of solutions(m, n)\in \mathbb{N}^2$$( m , n ) ∈ N 2 , to the inequality$$| \alpha^n - \beta^m | \leq x | α n - β m | ≤ x when x tends to