Unsolved Problems

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...

Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,...

Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$, the discrepancy $ D(N)=\max_{I\subseteq ...

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$, $ \sum_{1\leq k\leq N}...

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$...

Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subsete...

For any $0<\alpha<1$, let $ f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}). $ Does $f(\alpha,n)$ have an asymptotic dist...

Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function?...

Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log x)...

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1\leq k<l\leq ...

Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle. Determine $f...

Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)...

Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on ...

Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\leq k\leq n$. ...

Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $...

Is it true that, for every $k\geq 3$, there is a constant $c_k>0$ such that $ \mathrm{ex}(n,G_k) \ll n^{3/2-c_k}, $ where $G_k$ is the bipartite graph...

Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least ...

If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, ...

If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that $ \lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c. $ ...

We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$. Is there, f...

Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)...

Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?...

Determine the infimum and supremum of $ \lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert $ as $f\in \mathbb{R}[x]$ ranges over all non-consta...

Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is ...

Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of $ \lvert \{ z: \lvert f(z)\rvert < 1\}\rvert, $ as $f$ ranges...

Let $t>1$ be a rational number. Is $ \sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n} $ irrational, where $\tau(n)$ counts the d...

Is it true that if $a_1<a_2<\cdots$ is a sequence of integers with $ \liminf a_n^{1/2^n}>1 $ then $ \sum_{n=1}^\infty \frac{1}{a_na_{n+1}} $ is irrati...

A unitary divisor of $n$ is $d\mid n$ such that $(d,n/d)=1$. A number $n\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (...

Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$?...

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$? Or...

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$...

Let $k\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that $ \prod_{n\in I_i}n \equiv 1\pmod{p} $ for all $1\leq...

Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$. Is it true that $C(x)=x^{1-o(1)}$?...

Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$?...

Let $f(n)$ count the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\leq n^{o(\frac{1}{\...

How many solutions are there to $ \sigma(a)+\sigma(b)=\sigma(a+b) $ with $a+b\leq x$, where $\sigma$ is the sum of divisors function? Is it $\sim cx$ ...

Let $f(n)$ be the size of the largest subset $A\subseteq \{1,\ldots,n\}$ such that there are no three distinct elements $a,b,c\in A$ such that $a\mid ...