Unsolved Problems

Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i<k$. Estimate $n_k$....

Are there infinitely many primes $p$ such that $p=2^kq+1$ for some prime $q$ and $k\geq 0$? Or $p=2^k3^lq+1$?...

Let $G$ be a graph given by $n$ points in $\mathbb{R}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between t...

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?...

Let $f(n)$ be maximal such that, given any $n$ points in $\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $...

Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maxima...

For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\equiv 0\pmod{p}$. Is it true that there are infinitely many $p$ for which $f(p)=...

Let $A(x)$ count the number of composite $u<x$ such that $n!+1\equiv 0\pmod{u}$ for some $n$. Is it true that $A(x)\leq x^{o(1)}$?...

Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p ot\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does $ \lim \frac{\lvert S...

Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds. Any $r$-uniform hypergrap...

Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Est...

Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ ma...

Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. ...

Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Est...

Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in t...

Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances ...

Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d...

Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r...

Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic numb...

For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undef...

For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions....

Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$....

Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$. Is i...

Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there al...

If $1=d_1<\cdots<d_{\tau(n)}=n$ are the divisors of $n$, then let $\tau_\perp(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$. Is it true that...

If $u=\{u_1<u_2<\cdots\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i eq j$ and $\sum \frac{1}{u_i}<\infty$ then let $\{a_1<a_2<\cdot...

Let $A$ be an infinite sequence of integers such that every $n\in A+A$ is squarefree. How fast must $A$ grow?...

Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$....

The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rai...

Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of $ \prod_{1\leq k\leq n}p(k). $ Does $F(n)...

Let $r\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Is every large integer the sum of at most $r...

Let $ A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}. $ If $k\geq 2$, then does $A$ contain only finitely many $k$th powers...

Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is ...

Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other....

If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$. If $t,c\ge...

Let $1\leq d_1<d_2$ and $k\geq 3$. Does there exist an integer $r$ such that if $B=\{b_1<\cdots\}$ is a lacunary sequence of positive integers with $b...

A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a cove...

Let $f(z)$ be an entire function which is not a monomial. Let $ u(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\ma...

Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let $ E= \{ z : \lvert f(z)\rvert \le...

Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let $ A=\{ n \geq 1: f(n+1)< f(n)\}. $ If $\lver...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds. For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\...

Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd. Given any integer $m\geq 1$ does there exist...