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For a wide class of integer linear recurrence sequences 
$(u(n))_{n=1}^\infty $, we give an upper bound on the number of s-tuples 
$\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements 
$u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.
In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions 
$f_1, \ldots, f_n\in K(X)$, there are only finitely many elements 
$\alpha \in K$ such that 
$f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.
Let A be a commutative domain of characteristic 0 which is finitely generated over ℤ as a ℤ-algebra. Denote by A* the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the setwhere F(X, Y) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form XmYn - α or Xm - α Yn, with m, n ∈ ℤ⩾0, max(m, n) > 0, α ∈ K*. This result is a common generalisation of effective results of Evertse and Győry [12] on S-unit equations over finitely generated domains, of Bombieri and Gubler [5] on the equation F(x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem [20] on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.
\begin{equation*}\mathcal{C}:=\{ (x,y)\in (A^*)^2 | F(x,y)=0 \}\end{equation*}
Let 
$b\,>\,1$ be an integer. We prove that for almost all 
$n$ , the sum of the digits in base 
$b$ of the numerator of the Bernoulli number 
${{B}_{2n}}$ exceeds 
$c$ log $n$ , where 
$c\,:=\,c\left( b \right)\,>\,0$ is some constant depending on $b$ .
Let d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field 
. It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.
The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety 
, that with respect to the height are “very close” to a given subgroup Γ of finite rank of 
. Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.
In this paper we prove, for certain special classes of varieties
