Linear Forms in Logarithms and Applications

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