Unsolved Problems

Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any o...

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ su...

Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for a...

Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in...

Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacen...

Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $ \sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}. $ I...

Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} $ with...

Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}. $ Is i...

Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k...

Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b}+\frac{1}{c} $ with di...

For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with $ \frac{a}{b}=\frac{1}{n_1}+\cdots...

Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $ \frac{a}...

Let $\delta(N)$ be the minimal non-zero value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$. Is it t...

Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\fr...

Are there infinitely many solutions to $ \frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m}, $ where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are di...

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $ 0< \left\lvert \sum_{...

Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empt...

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that $ \sum_{n\in A}\frac{\delta...

Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$....

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?...

Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations ...

Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ ...

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?...

Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that $ f_{k,3}(x...

Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that $ \lim...

Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a eq b$ then $a+b mid ab$. Can $A$ be 'substantially more' than the odd numbers? W...

Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can $ \limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}} $ be?...

Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which can...

Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $...

Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prim...

Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d...

For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the ...

The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from...

Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that...

Let $A=\{a_1<\cdots<a_k\}$ be a finite set of positive integers and extend it to an infinite sequence $\overline{A}=\{a_1<a_2<\cdots \}$ by defining $...

With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i<j\leq n$. What ca...

Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq m...

Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that {UL} {LI} $A\backslash B$ is complete for any finite subset $B$ and {/LI} {LI} $A\bac...

For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that {UL} {LI} $A$ remains complete after ...

For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum...

Let $p(x)\in \mathbb{Q}[x]$. Is it true that $ A=\{ p(n)+1/n : n\in \mathbb{N}\} $ is strongly complete, in the sense that, for any finite set $B$, $ ...

Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?...

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset $ \{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloo...

Let $1\leq a_1<\cdots <a_k\leq n$ be integers such that all sums of the shape $\sum_{u\leq i\leq v}a_i$ are distinct. Let $f(n)$ be the maximal such $...

Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $ n=\sum_{u\leq i\leq v}a_i. $ Is there such a...

Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlie...

Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a s...

Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square...

Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that,...

How large is the largest prime factor of $n(n+1)$?...