Unsolved Problems

Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$?...

Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$, satisfies $ P\left(\binom{n}{k}\right)\g...

For $0\leq k\leq n$ write $ \binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let...

Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divi...

Can every integer $N\geq 2$ be written as $ N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)} $ for some $k\geq 2$ and $m\geq n+k$?...

Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ ...

Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}<p\leq n$ such that every...

Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$ satisf...

Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k...

Given $A\subseteq \mathbb{N}$ let $M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and sufficie...

Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divis...

Let $f_{\max}(n)$ be the largest $m$ such that $\phi(m)=n$, and $f_{\min}(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function. Inve...

Let $p_1<p_2<\cdots$ be a sequence of primes such that $p_{i+1}\equiv 1\pmod{p_i}$. Is it true that $ \lim_k p_k^{1/k}=\infty? $ Does there exist such...

Let $h(n)$ be the largest $\ell$ such that there is a sequence of primes $p_1<\cdots < p_\ell$ all dividing $n$ with $p_{i+1}\equiv 1\pmod{p_i}$. Let ...

Let $ f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right). $ {UL} {LI}Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)...

Let $g(n)$ be minimal such that for any $A\subseteq [2,\infty)\cap \mathbb{N}$ with $\lvert A\rvert =n$ and any set $I$ of $\max(A)$ consecutive integ...

Let $f(n)$ be minimal such that, for any $A=\{a_1,\ldots,a_n\}\subseteq [2,\infty)\cap\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\max(A)$ co...

Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain a...

Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Prov...

Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that $ \mathrm{ex}(n;G)\sim cn^\alpha? $ Must $\alp...

Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that $ f(n) \gg n^{1/2}? $ ...

As $n\to \infty$ ranges over integers $ \sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}. $ ...

Let $k\geq 2$. Does $ (n+k)!^2 \mid (2n)! $ for infinitely many $n$?...

Are there infinitely many pairs of integers $n eq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?...

Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m mid \binom{2n}{n}$ satisfies $ m\sim f...

Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a subgraph of chro...

Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 11$. Find the ...

Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv 1\pmod{p}$. I...

Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give g...

Let $h(n)$ be minimal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are mutually coprime. Does, for every prime $p$, the density $\delta_p$ of integers with...

What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$?...

We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n eq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X eq Y$. Let $A\subset \mathbb{N}$...

Fix some constant $C>0$ and let $n$ be large. Let $A\subseteq \{2,\ldots,n\}$ be such that $(a,b)=1$ for all $a eq b\in A$ and $\sum_{n\in A}\frac{1}{...

Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can onl...

Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert B\r...

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,n\}$ such that $a mid bc$ whenever $a,b,c\in A$ with $a eq b$ and $a eq c$. ...

Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $...

Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$. Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-...

Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $ g(n) > ...

Let $h(x)$ count the number of integers $1\leq a<b<x$ such that $(a,b)=1$ and $\sigma(a)=\sigma(b)$, where $\sigma$ is the sum of divisors function. I...

Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?...

Is it true that, for any $a\in\mathbb{Z}$, there are infinitely many $n$ such that $ \phi(n) \mid n+a? $ ...

Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edg...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that $ \limsup \frac{a_n...

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if $ \lvert A+A\rvert=(1+o(1))...

Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that $ \binom{n}{k}=a $ (with $1\leq k\leq n/2$) has exactly $t$ solution...

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ al...

Let $\epsilon>0$. Is there some $r\ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\geq 0$ and $n$ has at most $r$ prime...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ ar...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq x$. I...