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.

We study the complexity of the infinite word $u_{\beta }$ associated with the Rényi expansion of $1$ in an irrational base $\beta \>1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ\left(n\right)=n+1$. For $\beta$ such that $d_{\beta }\left(1\right)=t_1{t}_2\cdots t_m$ is finite we provide a simple description of the structure of special factors of the word $u_{\beta }$. When $t_m=1$ we show that $ℂ\left(n\right)=(m-1)n+1$. In the cases when $t_1=t_2=\cdots =t_{m-1}$ or $t_1\>max\{t_2,\cdots ,t_{m-1}\}$ we show that the first difference of the complexity function $ℂ(n+1)-ℂ\left(n\right)$ takes value in $\{m-1,m\}$ for every $n$, and consequently we determine...

We study the complexity of the infinite word uβ associated with the Rényi expansion of 1 in an irrational base β > 1. When β is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity C(n) = n + 1. For β such that dβ(1) = t1t2...tm is finite we provide a simple description of the structure of special factors of the word uβ. When tm=1 we show that C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or t1 > max{t2,...,tm-1} we show that the first difference of...

We describe the average behaviour of the Brjuno function Φ in the neighbourhood of any given point of the unit interval. In particular, we show that the Lebesgue set of Φ is the set of Brjuno numbers and we find the asymptotic behaviour of the modulus of continuity of the integral of Φ.

We exploit the properties of Legendre polynomials defined by the contour integral ${𝐏}_n\left(z\right)={\left(2\pi \mathrm{i}\right)}^{-1}\oint {(1-2tz+t^2)}^{-1/2}{t}^{-n-1}\mathrm{d}t,$ where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer $r$, a prime $p⩾5$ and $n=r{p}^2-1$, we have ${\sum }_{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv 0,1\text{or}-1(mod{p}^2)$, depending on the value of $r(mod6)$.