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The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and …

[Edit, Dec 6, 2019] I have a pleasure to inform that this problem was finally resolved in affirmative by T.Banakh, D.Spirito and S.Turek who proved the following Theorem. The Golomb space is topologi …

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 …

A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$. For a finite group $G$ denote by $sl(G)$ the smallest car …

I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive in …

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\ …

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 …

Problem. What is the smallest cardinality $d(n)$ of a set $A$ of integer numbers such that the difference set $A-A=\{a-b:a,b\in A\}$ contains $n$ consequtive integer numbers? It can be shown that $(1 …

Let $\mathsf C_n$ denotes the statement: for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for …

$\DeclareMathOperator\Aut{Aut}$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 …

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 …

I am writing a paper on the topological structure of the Golomb space (defined here) and arrived to the following question: Question 1. Is it true that for a number $a\in\mathbb N$ the equation $x^2+ …

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 $q$ …

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 …

Question 1. Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$? …