We show that $n^2+1$ is powerfull for $O\left(x^{2/5+ϵ}\right)$ integers $n\le x$ at most, thus answering a question of P. Ribenboim.

We prove: If $A\left(n\right)$ and $G\left(n\right)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA\left(n\right)/G\left(n\right)-(n-1)A(n-1)/G(n-1)$ $(n\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty$.

Every real number $x,0\<x\le 1$, has an essentially unique expansion as a Pierce series : $$x=\frac{1}{x^1}-\frac{1}{x^1{x}^2}+\frac{1}{x^1{x}^2{x}^3}-\cdots$$ where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $$y=\frac{1}{y^1}-\frac{1}{y^1{y}^2}+\frac{1}{y^1{y}^2{y}^3}+\cdots$$ where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

The tempered fundamental group of a $p$-adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between $\left(p^{\prime }\right)$ versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log...

Any sequence $x={\left(x_k\right)}_{k=1}^{\infty }$ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let $H_n\left(x\right)$ be the height of the tree generated by $x_1,\cdots ,x_n.$ Obviously $$\frac{logn}{log2}-1\le H_n\left(x\right)\le n-1.$$ If the sequences $x$ are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists $c$ > $0$ such that...

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_1+p_2+p_3=N$ where $N$ is an odd integer and the unknowns $p_i$ are prime numbers of the form $p_i=\left[n^{1/{\gamma }_i}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ < $\gamma$ < $1$. ...

We investigate the extremal function $f(u,n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v=a_1{a}_2...a_m$ of integers such that 1) $1\le a_i\le n$, 2) $a_i=a_j,i\ne j$ implies $|i-j|\ge k$ and 3) $v$ contains no subsequence of the type $u$. We prove that $f(u,n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $$f(u,n)=O\left(n{2}^{O\left(\alpha {\left(n\right)}^{\left|u\right|-4}\right)}\right).$$ Here $\left|u\right|$ is the length of $u$ and $\alpha \left(n\right)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and...