3,489 problems

Give an explicit profinite presentation of the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}_2/\mathbb{Q}_2)$.

If $A(x)$ counts integers satisfying the Sylow divisor condition, determine the constant $c$ in $A(x)/x=\exp(-(c+o(1))\sqrt{\log x}\log\log x)$.

Estimate the number $S(N)$ of distinct reciprocal subset sums $\sum_{n\in A}1/n$ with $A\subseteq\{1,\ldots,N\}$.

Define $\varphi_k(n)=\sum_{1\leq a\leq n,(a,n)=1}a^k$ and $\mathcal D_s=\{k\geq s:\varphi_s(n)\mid\varphi_k(n)\text{ for every }n\}$. Is $\mathcal D_1=\{1,3,15\}$?

Let $A=\{a_1<a_2<\cdots\}$ have positive lower logarithmic density. Must it contain a divisibility chain $a_{n_i}\mid a_{n_{i+1}}$ such that …

Given $\epsilon,\eta>0$, does some $k$ force the following? For any primes $p_1<\cdots<p_k<n^{1-\epsilon}$ and any choice of $(p_j-1)/2$ residue classes modulo each $p_j$, fewer th…

Can every sufficiently large integer $n$ be represented as $x^2+y^2-z^2$ with each of $x^2,y^2,z^2$ at most $n$?

Are there infinitely many $n$ such that $n-k^2$ is prime for every $k$ coprime to $n$ with $k^2<n$?

Let $x/2<y<x$, $C>1$, and let $d$ be the largest prime gap below $x$. Must $\pi(y+Cd)-\pi(y)\sim Cd/\log y$?

Can one choose a function $f$ and a number of colours $k$ so that every $k$-colouring of the integers contains a slowly growing sequence whose finite subset sums omit at least one…

If an additive function satisfies $\limsup_{p,k}f(p^k)/\log p^k=\infty$, must $\limsup_n(f(n+1)-f(n))/\log n=\infty$? Or even $\limsup_n f(n+1)/f(n)=\infty$?

For $A,B\subseteq[1,N]$, how many integers can have exactly one factorization $m=ab$ with $a\in A$ and $b\in B$?

How large can $A\subseteq[1,n]$ be if every ordered quadruple $a\leq b\leq c\leq d$ in $A$ with $abcd$ a square must satisfy $ad=bc$?

If $A$ is an additive basis of order 2 and its representation count tends to infinity, can $A$ be partitioned into two disjoint additive bases of order 2?

How large can $\frac1{\log N}\sum_{n\in A}1/n$ be when $A\subseteq[1,N]$ contains no relation $at=b$ whose multiplier $t$ has least prime factor greater than $a$?

For every $\epsilon>0$, is there a bounded $r$ such that integers of the form $2^k+n$, with $n$ having at most $r$ prime factors, have density at least $1-\epsilon$?

Is the largest subset $A\subseteq[1,N]$ for which $ab+1$ is never squarefree attained by the residue class $7\pmod{25}$?

For each constant $C>0$, are there infinitely many $a,b,n$ with $a+b>n+C\log n$ such that the denominator of $n!/(a!b!)$ has only bounded prime factors?

If $f_{\max}(n)$ and $f_{\min}(n)$ are the largest and smallest solutions of $\phi(m)=n$, how large can $f_{\max}(n)/f_{\min}(n)$ be for $n\leq x$?

For fixed $k$, is the density $d_k(p)$ of integers whose $k$th-smallest prime factor is $p$ unimodal as $p$ ranges over the primes?

For a finite abelian group $G$ of order $N$, let $f(N)$ be the smallest size of a random subset that generates every element as a subset sum with probability at least 1/2. Is…

Is there $\epsilon>0$ such that infinitely many $n$ have every prime $p\leq(2+\epsilon)\log n$ dividing $\prod_{1\leq i\leq\log n}(n+i)$?

Can one find a function $f(r)\to\infty$ such that infinitely many $n$ admit $a_1+a_2>n+f(r)\log n$ while $a_1!a_2!$ divides $n!2^n3^n\cdots p_r^n$?

Are there only finitely many identities $\prod_i {2m_i\choose m_i}=\prod_j {2n_j\choose n_j}$ when all the $m_i$ and $n_j$ are distinct?

Call $[u,v]$ bad when the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with exponent greater than 1. Is the count of integers up to $x$ lying in a bad interval asymptot…