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We define the lower density of a set $A\subseteq \mathbb{N}$ by $$ \operatorname{ld}(A) \ := \ \liminf_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}. $$ For $A,B\subseteq \mathbb{N}$, we set $$ A + …

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a su …

Let us call a positive integer $n\in\mathbb{N}$ consecutively summable if there are positive integers $m, k < n$ such that $$n=\sum_{i=0}^k (m+i).$$For $A\subseteq \mathbb{N}$ we set the lower density …

Let $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum_{j\in S}j^{-1}=k$. …

Let $\mathbb{N}$ denote the set of positive integers. If $A\subseteq \mathbb{N}$ is finite, we say that $A$ is reciprocally summable to $1$ ("rs1") if $\sum_{a\in A} \frac{1}{a} = 1$. If $A\subseteq …

Let $\mathbb{N}$ denote the set of the positive integers. We consider the following function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{Q}$: $$f(a,b)=\frac{a^2+b^2}{1+ab} \text{ for all } a,b\in\mathbb …

Let $P = \{n^k: n,k\in\mathbb{N}\setminus\{0,1\}\}$ denote the set of powers. For any $n,r\in\mathbb{N}$ we set $B_r(n)=\{m\in\mathbb{N}: |m-n| \leq r\}$. Is there a "global" constant $K\in\mathbb{N …

Given positive integers $a,b\in\mathbb{N}$ with ${\text gcd}(a,b) = 1$, and given a positive integer $d$, are there necessarily positive integers $m,n$ such that $d \;| \; (2^ma + 2^nb)$?

The number $2$ has the interesting property that whenever $n>1$ is an integer, then $n \nmid (2^n-1)$. (It's a good exercise to prove this statement.) Let's call a positive integer $b$ $2$-like if fo …

Let $\mathbb{N}$ denote the set of positive integers, and let ${\bf P}\subseteq {\mathbb N}$ be the set of prime numbers. For $a\in \mathbb{N}$ let $$p(a) = \{p\in {\bf P}: (\exists k\in\mathbb{N})k\c …

We say that a set $A\subseteq \mathbb{N}$ has positive measure if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$ For $b\in\mathbb{N}$ with $b>1$ we consider the sets $$S_b := \bi …

Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R\subseteq \mathbb{N}^3$ by $$ R = \{(x,y,z) \in \mathbb{N}^3: \exists n\in \mathbb{N}: 1< n \leq \max\{x,y,z\} \land \exi …

Is this statement true for all positive integers $n\in\mathbb{N}$? For all $\varepsilon >0$ there are prime numbers $p,q$ such that $|\frac{p}{q} - n| < \varepsilon$.

Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two …

Is there $q\in\mathbb{Q}$ with $q>1$ and the following property? There are infinitely many $n\in\mathbb{N}$ such that there are no primes in $[n,\lfloor q\cdot n\rfloor]$.