Unsolved Problems

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 $n(n+1)$?...

Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ a...

Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$....

Show that the equation $ n! = a_1!a_2!\cdots a_k!, $ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions....

For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots <a_k=m$ with $a_1!\cdots a_k!$ a square....

Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?...

Is there some absolute constant $C>0$ such that $ \sum_{p\leq n}1_{p mid \binom{2n}{n}}\frac{1}{p}\leq C $ for all $n$ (where the summation is restric...

We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count ...

Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can...

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \prod_{0\leq i\leq k}(p^2+i) $ is $p$?...

Let $ F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m), $ where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all s...

Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $ \binom{21}{2}=2\cdot 3\cdot 5\cdot 7. $...

Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?...

Can one classify all solutions of $ \prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there...

Is it true that for every $n\geq 1$ there is a $k$ such that $ n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)? $ ...

Let $f(n)$ be the minimal $m$ such that $ n! = a_1\cdots a_k $ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $ f(n)-...

Let $f(n)$ denote the minimal $m\geq 1$ such that $ n! = a_1\cdots a_t $ with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?...

Let $t_k(n)$ denote the least $m$ such that $ n\mid m(m+1)(m+2)\cdots (m+k-1). $ Is it true that $ \sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c} $ fo...