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Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0 …

$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\A …

There is a quite elementary proof that uses only the fact that a polynomial of degree $n$ has at most $n$ roots in a field. So, let $F$ be a field and $G$ be a finite subgroup of the multiplicative gr …

Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the ``Euclidean'' plane $F^2$ endowed with the …

This question is follow up of this MO-post. First let us recall the necessary definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0 …

Let $(F_k)_{k=0}^\infty$ be the classical Fibonacci sequence, defined by the recursive formula $F_{k+1}=F_k+F_{k-1}$ where $F_0=0$ and $F_1=1$. For every $n\in\mathbb N$ let $\pi(n)$ be the smallest p …

The following question was motivated by this MO-post. I hope that the answer should be known to experts (because of very simple formulation)... Problem. Let $n\ge 2$. Is the set of complex numbers $\ …

I need to cite the classical Zsigmondy Theorem, which was proved in 1892. Is there any modern reference to this theorem? I mean some standard textbook in Number Theory containing this theorem together …

Definition. A prime number $p$ is called strange if there exists $k>1$ such that each prime divisior of $p^k-1$ divides $p-1$. Example 3. The prime number $p=3$ is strange as $3^2-1=8$ has the same pr …

An prime number $p$ is called pathological if there exists a prime number $q\ne p$ such that for every $n\in\mathbb N$ the number $2^n-1$ is divisible by $p$ if and only if $2^n-1$ is divisible by $q$ …

Let $\Pi$ be the set of odd prime numbers and let $\mathcal P(\Pi)$ be the Boolean algebra of subsets of $\Pi$. For a number $x$ denote by $\Pi(x)$ the set of odd prime divisors of $x$. Problem. …

I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in Wikipedia. Theorem (Gauss). Let $p$ be a prime number, $k\in\mathbb N$ and $\m …

For a prime number $p$, the $p$-adic topology on the set $\omega$ of non-negative integers is generated by the base consisting of the arithmetic progressions $x+p^n\omega:=\{x+p^ny:y\in\omega\}$ where …

Writing down the details of the argument of Ilya Bogdanov, we can obtain the following upper bound: Theorem. $f(n)\le\frac83n-1$ for every $n\ge 2$. Proof. If $n=3k+1$ or $n=3k+2$, then following th …

In this preprint (written jointly with Alex Ravsky) we prove the following partial answers to this problem. First some definitions. A non-empty subset $D$ of an Abelian group is called decomposable if …