The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb{N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $min\{x,y,z\}>1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation$$\begin{array}{cc}\hfill {\Phi }_n(x,y)& =x^3+(n^8+2n^6-3n^5+3n^4-4n^3+5n^2-3n+3)x^{2}y\hfill \\ & -(n^3-2)n^{2}x{y}^2-y^3=\pm 1,\hfill \end{array}$$for $n\ge 0$.

We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a{13}^b{17}^c=2^m{y}^n$, where $a,b,c,m\ge 0,n\ge 3$ and $x,y\ge 1$ are unknown integers with $gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.

We present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.We call a complex (quasiprojective) surface of hyperbolic type, iff – after removing finitely many points and/or curves – the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of...

On calcule par des méthodes arithmétiques le groupe de Brauer non ramifié des espaces homogènes de groupes algébriques linéaires sur différents corps. Les formules obtenues font intervenir l’hypercohomologie de complexes de groupes de type multiplicatif.

La complexité d’une suite infinie est définie comme la fonction qui compte le nombre de facteurs de longueur $k$ dans cette suite. Nous prouvons ici que la complexité des suites de Rudin-Shapiro généralisées (qui comptent les occurrences de certains facteurs dans les développements binaires d’entiers) est ultimement affine.

The aim of this paper is to evaluate the growth order of the complexity function (in rectangles) for two-dimensional sequences generated by a linear cellular automaton with coefficients in $\mathbb{Z}/l\mathbb{Z}$, and polynomial initial condition. We prove that the complexity function is quadratic when l is a prime and that it increases with respect to the number of distinct prime factors of l.