Asks whether $\sum_n 2^{-a_n}$ is transcendental for every increasing integer sequence $a_n$ with $\limsup a_n/n=\infty$.

• Number theory
• Irrationality

Asks whether $\sum_n \phi(n)/2^n$ is irrational, where $\phi$ is Euler's totient function.

• Number theory
• Irrationality

Asks whether $\sum_n p_n/2^n$ is irrational, where $p_n$ is the $n$th prime.

• Number theory
• Irrationality

Asks whether $\sum_{n\in A}\frac{1}{2^n-1}$ is irrational for every infinite set $A\subseteq\mathbb{N}$.

• Irrationality

Asks for which $n$ the number $n/2^n$ is a sum of distinct terms $k/2^k$, and whether some rational has uncountably many such sums.

• Number theory

Asks whether $a_n=2^{2^n}$ makes $\sum 1/b_n$ irrational for every sequence of positive integers $b_n$ with $b_n/a_n\to1$, and whether every $a_n$ with this property must satisfy $a_n^{1/n}\to\infty$.

• Irrationality

Asks whether $a_n=2^n$ or $a_n=n!$ makes $\sum 1/(a_n+b_n)$ irrational for every bounded sequence of nonzero integers $b_n$ with $a_n+b_n\ne0$. The case $a_n=2^n$ is answered no.

• Irrationality

Asks how fast an increasing integer sequence $a_n$ can grow if $\sum 1/a_n$ and $\sum 1/(a_n-1)$ are both rational.

• Irrationality

Asks whether $\sum_n 1/\operatorname{lcm}(a_1,\ldots,a_n)$ is irrational, where $a_1<a_2<\cdots$ are the integers whose prime factors lie in a fixed finite set of at least two primes.

• Irrationality

Asks for an explicit description and the growth rate of the greedy sequence that starts with $0$ and $n$ and avoids three-term arithmetic progressions.

• Additive combinatorics
• Arithmetic progressions

Asks for the largest number of subsets of $\{1,\ldots,N\}$ whose pairwise intersections are all non-empty arithmetic progressions. The exact value is open, while Szabó's linear-error question, whether it is $N^2/2+O(N)$, has a Lean proof that Conjectures.io accepted.

• Additive combinatorics
• Arithmetic progressions

Asks whether there is a covering system of congruences whose moduli are all of the form $p-1$ for primes $p\ge5$.

• Number theory
• Covering systems

Asks whether a group can be covered by more than one coset, the cosets having different sizes, so that each element lies in exactly one of them.

• Group theory
• Covering systems

Asks whether some Fibonacci-type sequence has all terms composite while no integer shares a factor with every term.

• Number theory
• Covering systems

Asks whether the greedy algorithm for unit fractions, which always takes the least allowed denominator, terminates for every rational in $(0,1)$ with odd denominator when only odd denominators are allowed.

• Number theory
• Unit fractions

Asks whether every representation $1=1/n_1+\cdots+1/n_k$ with $1<n_1<\cdots<n_k$ has two consecutive denominators differing by at least $3$.

• Number theory
• Unit fractions

Asks whether only finitely many pairs of integer intervals $I_1,I_2$ satisfy $\sum_{n\in I_1}\frac{1}{n}+\sum_{n\in I_2}\frac{1}{n}\in\mathbb{N}$.

• Number theory
• Unit fractions

Asks whether $(a_n,L_n)=1$ and $(a_n,L_n)>1$ each hold for infinitely many $n$, where $L_n=\mathrm{lcm}(1,\ldots,n)$ and $\sum_{k\le n}1/k=a_n/L_n$.

• Number theory
• Unit fractions

Asks for the growth of $v(k)$, the least integer greater than $1$ that is not a denominator in any representation of $1$ as a sum of $k$ distinct unit fractions.

• Number theory
• Unit fractions

Asks whether $k(N)-(e-1)N\to\infty$, where $k(N)$ is the least number of distinct unit fractions, all with denominators at least $N$, that sum to $1$.

• Number theory
• Unit fractions

Asks whether there are two finite sets of primes whose sums of reciprocals have product $1$.

• Number theory
• Unit fractions

Asks whether the least non-zero distance $\delta(N)$ from $1$ to a sum $\sum_{n\in A}1/n$ with $A\subseteq\{1,\ldots,N\}$ equals $e^{-(c+o(1))N}$ for some constant $c\in(0,1)$.

• Number theory
• Unit fractions

Asks whether, for some $c>0$, every large enough finite multiset of positive integers with reciprocal sum above $K$ has a subset whose reciprocal sum lies in $(1-e^{-cK},1]$.

• Number theory
• Unit fractions

Asks whether infinitely many sums of reciprocals of distinct primes equal $1-1/m$ for an integer $m\ge2$.

• Number theory
• Unit fractions

Asks whether, for some $c>0$ and every $n$, signs $\delta_k\in\{-1,0,1\}$ can be chosen with $0<|\sum_{k\le n}\delta_k/k|<c/2^n$.

• Number theory
• Unit fractions