Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D,p)$, and also prove that if the equation $U^2-D{V}^2=-1$ has integer solutions $(U,V)$, the least solution $(u_1,v_1)$ of the equation $u^2-p{v}^2=1$ satisfies $p\nmid v_1$, and $D>C\left(p\right)$, where $C\left(p\right)$ is an effectively computable constant...

Let p(z) be a polynomial of the form $p\left(z\right)={\sum }_{j=0}^n{a}_j{z}^j$, $a_j\in -1,1$. We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

In this note we provide a direct and simple proof of a result previously obtained by Astier stating that the class of spaces of orderings for which the pp conjecture holds true is closed under sheaves over Boolean spaces.

Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $char\left(K\right)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $Jac\left(C\right)$ has semi-stable reduction over $R$. Embed $C$ in $Jac\left(C\right)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion...

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}:y^q=x^p+a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}\left(\mathbb{Q}\right)$ (resp. $J_{q,p,a}\left(\mathbb{Q}\right)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over ${}_l$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...

The paper deals with several criteria for the transcendence of infinite products of the form ${\prod }_{n=1}^{\infty }\left[b_n{\alpha }^{a_n}\right]/b_n{\alpha }^{a_n}$ where $\alpha >1$ is a positive algebraic number having a conjugate ${\alpha }^{*}$ such that $\alpha \ne |{\alpha }^{*}|>1$, ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P. Corvaja, U. Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta...