Asks whether some graph on the ordinal $\omega_2^2$ has chromatic number $\aleph_2$ while every subgraph of smaller type is countably colorable, and asks the same with chromatic number $\aleph_1$.

• Graph theory
• Chromatic number

Asks for the least $x$ such that a positive density of $n$ have each of $n+1,\ldots,n+k$ divisible by a prime $\le x$, and whether it is at least $k^{1-o(1)}$. It is the inverse of the covering function of Problem 687 and is open, with lower bounds of order $\sqrt{k}$ refereed and $\sqrt{k}\log k$ resting on a pending claim.

• Number theory

Asks whether, for every $r$, there is a $k$ such that the product of all integers in any $r$ disjoint intervals of consecutive integers, each of length at least $k$, is never a perfect power.

• Number theory

Asks whether, for $k_1\geq k_2\geq 3$, only finitely many disjoint blocks of $k_1$ and $k_2$ consecutive integers have products with the same prime factors.

• Number theory

Asks whether infinitely many prime gaps $(p_r,p_{r+1})$ contain at least two integers whose prime factors are all smaller than $p_{r+1}-p_r$.

• Number theory

Asks whether the largest divisor of $n(n+1)$ built only from the primes 2 and 3 exceeds every fixed multiple of $n\log n$ for suitable $n$.

• Number theory

Estimates $h_t(d)$, the least number of edges forcing a graph of maximum degree at most $d$ to have two edges at distance at least $t$. Open: it is known exactly for $t=1$ and $t=2$ and at $h_3(3)=23$, lies between $0.629^td^t$ and $\frac32d^t+1$ in general, and 2026 preprints treat $t=3$.

• Graph theory

Asks whether the powerful part of $n(n+1)\cdots(n+\ell)$ is below $n^{2+\epsilon}$ for every $\epsilon>0$ once $n$ is large and, for $\ell\geq 2$, whether its ratio to $n^2$ is unbounded and whether its ratio to $n^{\ell+1}$ tends to zero.

• Number theory
• Powerful numbers

Asks whether $2^n\pm 1$ and $n!\pm 1$ are powerful numbers for only finitely many $n$.

• Number theory
• Powerful numbers

Asks whether only finitely many runs of three consecutive powerful numbers (integers divisible by the square of each of their prime factors) are equally spaced.

• Number theory
• Powerful numbers

Asks whether, for $r\geq 4$, a sum of $r-2$ coprime $r$-powerful numbers can be $r$-powerful and if so only finitely often, and whether $a+b=c$ has infinitely many coprime 3-powerful solutions. The last is answered yes (Nitaj 1995), every $r\geq 6$ has infinitely many solutions, and $r=4$ is open.

• Number theory
• Powerful numbers

Asks, for $r\geq 3$, whether infinitely many integers are not a sum of at most $r$ numbers divisible by the $r$th power of each of their prime factors, and whether the integers that are such sums have density $0$.

• Number theory
• Powerful numbers

Asks for estimates of the number $h(n)$ of powerful integers in $[n^2,(n+1)^2)$, in particular whether some $c>0$ gives $h(n)<(\log n)^{c+o(1)}$ for all $n$ and $h(n)>(\log n)^{c-o(1)}$ for infinitely many $n$.

• Number theory
• Powerful numbers

Asks whether the number of ways to write $n$ as a sum of two powerful numbers is $n^{o(1)}$.

• Number theory
• Powerful numbers

Asks for an estimate of the longest run of consecutive integers up to $x$ with all divisor counts distinct, and whether, for some constant $C>0$, every interval $[x,x+(\log x)^C]$ with $x$ large repeats a divisor count.

• Number theory
• Divisors

Asks whether the complement of every sum-free set $S\subseteq\mathbb R$ contains a set $A$ of size continuum with $A+A$ also outside $S$. Open, with AlphaProof's Lean proof of the Sidon case pending as a claim and a thread comment arguing the Baire case.

• Ramsey theory

Asks whether $f(n)=\sum_{p<n}1/(n-p)$, summed over primes $p<n$, has $\liminf f(n)=1$ and $\limsup f(n)=\infty$, and whether $f(n)=o(\log\log n)$.

• Number theory
• Primes

Asks whether every real sequence $1<a_1<a_2<\cdots$ whose products $\prod_i a_i^{k_i}$ for different exponent tuples differ by at least $1$ satisfies $\#\{a_i\le x\}\le\pi(x)$. The quantifier over $x$ is implicit, and the readings for every $x$ and for all large $x$ are separate parts.

• Number theory

Asks how large the measure of a set in a disc of radius $r$ can be if no two of its points are at an integer distance.

• Geometry
• Distances

For the sequence with $a_0=0$, $a_1=1$ and each next term the least $n$ for which $a_i+a_j\le n$ has fewer than $n$ solutions, asks whether the number of solutions of $a_i+a_j\le x$ is $x+O(x^{1/4+o(1)})$.

• Number theory

Asks whether every set $A$ of density zero has a preimage $s^{-1}(A)$ of density zero, where $s(n)=\sigma(n)-n$ is the sum of proper divisors. Known results cover the primes, sums of two squares, integers with unusually many prime factors, palindromes, and sparse and missing-digit sets.

• Number theory

Asks to estimate the largest possible gap between the two highest distance multiplicities determined by a set of $n$ points in the plane.

• Geometry
• Distances

Asks for an estimate of the least $n$ such that every run of $n$ consecutive integers above $k$ contains one divisible by a prime greater than $k$; open, between a Rankin-type prime-gap lower bound and $k/\log k$ times iterated logarithms.

• Number theory

Asks for an estimate of the greatest $k$ such that, for some $m\le n$, each of $m+1,\ldots,m+k$ is divisible by a prime greater than $k$, and whether $\log k(n)\le(\log n)^{1/2+o(1)}$. Both are open. A published upper bound is $k(n)\ll n^{2/5}$; an author-recorded exponent-pair deduction gives $k(n)\ll_\varepsilon n^{38/97+\varepsilon}$.

• Number theory

Asks to estimate $f(n)$, the size of the largest dissociated subset guaranteed in any set of $n$ reals, and in particular whether $f(n)\ge\lfloor\log_2 n\rfloor$.

• Number theory