13 problems

Let $A\subseteq \{1,\ldots,N\}$ be such that there is no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of…

A set $A\subseteq\mathbb N$ is primitive if no member of $A$ divides another. Prove that for every primitive set $A$ whose elements are all at least $2$, …

Let $A=\{a_1<a_2<\cdots\}$ have positive lower logarithmic density. Must $A$ contain a chain $a_{n_1}\mid a_{n_2}\mid\cdots$ such that …

Is it true that, for every $x$, if $A\subseteq [x,\infty)$ is a primitive set of integers—meaning that whenever $a,b\in A$ and $a\mid b$, the integers $a$ and $b$ are associated, h…

Is there a necessary and sufficient condition for a sequence of integers $b_1<b_2<\cdots$ that ensures there exists a primitive sequence $a_1<a_2<\cdots$ (i.e. no element divides a…

Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the same set for both players) such that no…

Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let…

Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have $\lvert kx -y\rvert \geq 1.$ Does this imply that $A$ is spa…

Banks–Martin conjecture analogue. For each $q\geq 5$, the inequalities

Let $U(z)$ and $d6d1_1(z)$ be the functions defined in the paper for $z\in(0,2)$. Equality conjecture. … This is one of the paper's open questions concerning the relationship betwe…

Let $\Omega(n)$ be the number of prime factors of $n$, counted with multiplicity, and let $\mathbb{N}_k=\{n:\Omega(n)=k\}$. For a set of odd primes $\mathcal Q$, write…

For $x\ge 1$, let $[x,\infty)$ denote the integers at least $x$, and let $f(A)=\sum_{a\in A}1/(a\log a)$ for a primitive set $A$. Erdős–Sárközy–Szemerédi conjecture. … This questio…

Banks–Martin conjecture. The sequence of sums is strictly decreasing: