53 problems

For every integer $k\geq 2$ and every strictly increasing sequence of natural numbers $1<s(0)<\cdots<s(k-1)$ satisfying $\sum_{i=0}^{k-1}\frac{1}{s(i)}=1$, the maximum consecutive…

Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$. Is it true that,…

For every natural number $a\geq 1$, there exists a natural number $b>a$ such that the denominator, in lowest terms, of … is strictly smaller than the denominator, in lowest terms,…

Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that…

Define $u_0=1$ and $u_{i+1}=u_i(u_i+1)$, and define the Vardi constant … For every sequence $a:\mathbb N\to\mathbb N$ such that $a_i>0$ for all $i$, $a$ is strictly increasing,…

For a finite set $S\subset\mathbb{N}$, define its Egyptian sum by $E(S)=\sum_{m\in S}1/m$. It is an underapproximation of $x\in\mathbb{R}$ if every $m\in S$ is positive and…

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?

For every finite coloring of the integers, there exist distinct nonzero integers $a,b,c$ of the same color such that …

Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$.

For each natural number $n\geq1$, let $m(n)$ be the least natural number $m$ such that … and define … Then …

For a finite set $A\subseteq\mathbb N$, let … For natural numbers $N$ and $k$, say that there are $k$ disjoint unit decompositions in $\{1,\ldots,N\}$ if there exist pairwise disjo…

Does there exist a lacunary sequence $A=(A_n)_{n\in\mathbb N}$ of natural numbers and real numbers $u<v$ such that every rational number in the open interval $(u,v)$ is a finite su…

Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b mid ab$. Can $A$ be 'substantially more' than the odd numbers? What if $a,b\in A$ with…

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that $\sum_{n\in A}\frac{\delta_n}{n}=0$ and…

Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that…

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with…

If $A\subseteq\mathbb{N}$ is finite, $0\notin A$, $1\notin A$, and … is there always a partition $A=A_1\sqcup A_2$ into disjoint finite subsets such that …

Are there infinitely many solutions to $\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},$ where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?

Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some…

Let $\delta(N)$ be the minimal non-zero value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$. Is it true that…

For every $\alpha>0$, is there $B(\alpha)$ such that whenever $A\subseteq\{1,\ldots,N\}$ has $|A|\geq\alpha N$, some subset $S\subseteq A$ satisfies $\sum_{n\in S}1/n=a/b$ with…

How many integers can be represented as $\sum_{k=1}^N\epsilon_k/k$ with each $\epsilon_k\in\{0,1\}$? Is the number $o(\log N)$?

For $N≥1$, let $S_N$ be the set of integers representable as sums of distinct unit fractions with denominators at most $N$. What is the least positive integer outside $S_N$? Is…

Do there exist finite sets of primes $P$ and $Q$ such that …

For integers $1≤a<b$, let $D(a,b)$ be the least possible largest denominator in a representation of $a/b$ as a sum of distinct unit fractions, and let $D(b)=\max_{1≤a<b}D(a,b)$. Is…