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We call an integer $k\geq 1$ good if for all $q\in\mathbb{Q}$ there are $a_1,\ldots, a_k\in \mathbb{Q}$ such that $$q = \prod_{i=1}^k a_i \cdot\big(\sum_{i=1}^k a_i\big).$$ Euler showed that $k=3$ is …

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$ are absurdly high, namely $$(1544768 …

Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to Benford's law for the first digit in every base $b\geq 2$? Precise formulation of what it means for a set $T\subseteq \omega$ to "adhere to …

Motivation. In a coin game, a player flips all their coins every turn, starting with just one coin. If the coins all land heads then the game stops; otherwise, the number of coins is doubled for the f …

The "mortal matrix" problem: Given a set of $n\times n$-matrices with integer entries, decide whether they can be multiplied, in any order and possibly with repetition, to give the $0$-matrix. If I re …

Suppose we are given $A \subseteq \mathbb{N}$ with $\lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0$. For $k\in \mathbb{N}, k\geq 2$ we set $$M_A(k) = \{a\in A: ka \in A\}.$$ Does there exist …

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 …

For graph theory problems, I find this the most comprehensive resource: http://www.math.ucsd.edu/~erdosproblems/

Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors: $(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$, where $a,b$ are g …

We call a map $f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ an additive grid if for all $x,y \in {\mathbb Z}$ we have that $f(x,y)$ is the sum of the neighboring values, that is, $$f(x,y) = f(x-1, …

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 …

The starting point for this question is the following (false) statement $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$ Given a polynomial function $p:\mathbb{N} \to \mathbb{N …

The question whether the set $P\subseteq \mathbb{N}$ of perfect numbers is infinite, is famously open. I would think that everybody believes the statement below - but has it been proved? $$\mu(P) := …

For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum_{s\in S} s=\frac{p}{q}$, see this article by Paul Erdös and Sherman Stein …

We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$ Given $A,B\subseteq \mathbb{N}$ we set $A\cdot B = \{a\cdot b: a\i …