Unsolved Problems

If $ n! = \prod_i p_i^{k_i} $ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. Prove that there...

Are there infinitely many $n$ such that if $ n(n+1) = \prod_i p_i^{k_i} $ is the factorisation into distinct primes then all exponents $k_i$ are disti...

Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any...

Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\a...

Is there a graph $G$ with vertex set $\omega_2^2$ and chromatic number $\aleph_2$ such that every subgraph whose vertices have a lesser type has chrom...

Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$. Is it true that, for $k\geq 4$, $ f_k(n) \...

Let $\alpha,\beta\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)<n^{\alpha}$ and $...

Let $k\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where $ n+1,n+2,\ldots,n+k $ are all divisib...

Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k...

Let $k_1\geq k_2\geq 3$. Are there only finitely many $n_2\geq n_1+k_1$ such that $ \prod_{1\leq i\leq k_1}(n_1+i)\textrm{ and }\prod_{1\leq j\leq k_2...

Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r<n<p_{r+1}$ all of whose prime factors are $<p_{r+1}-p_r...

If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that $ \limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty? $ ...

Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\leq d$ contains two edges whose shortest path between them ...

For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $ Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}. $ Is it true t...

Are $ 2^n\pm 1 $ and $ n!\pm 1 $ powerful (i.e. if $p\mid m$ then $p^2\mid m$) for only finitely many $n$?...

Let $A=\{n_1<n_2<\cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions o...

Let $r\geq 2$. An $r$-powerful number $n$ is one such that if $p\mid n$ then $p^r\mid n$. If $r\geq 4$ then can the sum of $r-2$ coprime $r$-powerful ...

Let $r\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Are there infinitely many integers which are...

Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some cons...

Let $A$ be the set of powerful numbers (if $p\mid n$ then $p^2\mid n$). Is it true that $ 1_A\ast 1_A(n)=n^{o(1)} $ for every $n$?", "difficulty":...

A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number. Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$...

Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the ...

Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\cdots$ such that $a_n<f(n)$ for infi...

Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality continuu...

Let $ f(n) = \sum_{p<n}\frac{1}{n-p}. $ Is it true that $ \liminf f(n)=1 $ and $ \limsup f(n)=\infty? $ Is it true that $f(n)=o(\log\log n)$ for all $...

Let $1<a_1<\cdots$ be a sequence of real numbers such that $ \left\lvert \prod_i a_i^{k_i}-\prod_j a_j^{\ell_j}\right\rvert \geq 1 $ for every distinc...

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $ \lvert x_{n+1}-x_n\rvert \ll 1? $ ...

Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert ot\in \...

Let $1=a_1<a_2<\cdots$ be the sequence of integers defined by $a_1=1$ and $a_{k+1}$ is the smallest integer $n$ for which the number of solutions to $...

Let $ s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d $ be the sum of proper divisors function. If $A\subset \mathbb{N}$ has density $0$ then $s^{-1...

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $ \delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|. $ Let $h(n...

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the nu...

Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such t...

Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$....

Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers $ m+1,\ldots,m+k $ are divisible by at least one prime $...

Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\rvert\geq k$ w...

Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density?...

Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic $ Q(x)=\frac{6}{\p...

Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive...

Let $p(a,d)$ be the least prime congruent to $a\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$, $ p(a,d) > (1+c)\phi(d)\log ...

Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?...

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq ...

Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant ...

Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$ ...

Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k eq 2^l$ for any $l\geq 1$) such that the leading coefficien...

Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to $ n=p_1^k+\cdots+p_k^k, $ where the $p_i$ are prime numbers. Is it true that $\limsup...

Let $n\geq 2$ and $\pi(n)<k\leq n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ t...

Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?...

For any fixed $k\geq 3$, $ R(k,n) \gg \frac{n^{k-1}}{(\log n)^c} $ for some constant $c=c(k)>0$....

Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let $ A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert, $ where $e(x...