Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....

We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in [10]. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around $5.{10}^4$ instead of ${10}^8$.

Let $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)\in {ℝ}^3$. Suppose that $1,{\theta }_1,{\theta }_2,{\theta }_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$\begin{array}{cc}\hfill \alpha \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; sup}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \},\hfill \\ \hfill \beta \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; inf}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \}\hfill \end{array}$$ we prove $$\beta \left(\Theta \right)\ge \frac{1}{2}\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}+\sqrt{{\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}\right)}^2+\frac{4\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}}\right)\alpha \left(\Theta \right).$$

Let $\xi$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n\left(\xi \right)$ and $w_n^{*}\left(\xi \right)$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi$ by quadratic surds.

On montre que les exposants de Lyapunov de l’algorithme de Jacobi-Perron, en dimension $d$ quelconque, sont tous différents et que la somme des exposants extrêmes est strictement positive. En particulier, si $d=2$, le deuxième exposant est strictement négatif.

In 1955, Roth established that if $\xi$ is an irrational number such that there are a positive real number $\epsilon$ and infinitely many rational numbers $p/q$ with $q\ge 1$ and $|\xi -p/q|\&lt;q^{-2-\epsilon }$, then $\xi$ is transcendental. A few years later, Cugiani obtained the same conclusion with $\epsilon$ replaced by a function $q\mapsto \epsilon \left(q\right)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi -p/q|\&lt;q^{-2-\epsilon \left(q\right)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous...

We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between $\surd n-∜n{\left(logn\right)}^{1/7}$ and $\surd n+∜n{\left(logn\right)}^{1/7}$.

Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$-Fibonacci and $k$-Lucas numbers which are Fermat numbers. Some more general results are given.