Asks for the size of the largest $A\subseteq\{1,\ldots,N\}$ with signs $\delta_n=\pm1$ such that $\sum_{n\in A}\delta_n/n=0$ but no proper non-empty subset of $A$ gives zero.

• Number theory
• Unit fractions

Asks for the order of growth of the number of ways to write $n$ as a sum of $k$ many $k$th powers, and whether it exceeds $n^c$ for some fixed $c>0$ and infinitely many $n$.

• Number theory
• Powers

Asks whether $f_{k,k}(x)\gg_\epsilon x^{1-\epsilon}$ and, for $m<k$, whether $f_{k,m}(x)\gg x^{m/k}$, where $f_{k,m}(x)$ counts the integers up to $x$ that are sums of $m$ nonnegative $k$th powers.

• Number theory
• Powers

Asks whether some polynomial $f\in\mathbb{Z}[x]$ makes all the sums $f(a)+f(b)$, with $a<b$ nonnegative integers, distinct.

• Number theory
• Powers
• Sidon sets

Asks whether, for $k\ge3$, the number of integers up to $x$ that are sums of three nonnegative $k$th powers is at least a constant times $x^{3/k}$.

• Number theory
• Powers

Asks how large $\limsup_{N\to\infty}|A\cap\{1,\ldots,N\}|/N^{1/2}$ can be for an infinite Sidon set $A\subseteq\mathbb{N}$.

• Number theory
• Sidon sets

Asks which conditions on a set of integers ensure that the differences occurring infinitely often in it have bounded gaps.

• Number theory

Asks for the best $f(n)$ such that every $n$ is a sum of two integers with no prime factor larger than $f(n)$; open. Erdős asked whether $n^{\epsilon}$ suffices, which is unsettled, and whether $n^{1/3}$ does, which Balog's bound answers yes.

• Number theory

Asks for a characterization of the pairs of sets $A,B$ of integers with positive density such that $d(A+B)=d(A)+d(B)$.

• Number theory
• Additive combinatorics

Asks when a basis has a restricted order, defined with distinct summands only, and whether that order is bounded in terms of the ordinary order.

• Number theory
• Additive bases

Asks for the growth rate of the greedy Sidon sequence, and whether its counting function is $\gg N^{1/2-\epsilon}$ for every $\epsilon>0$.

• Number theory
• Additive combinatorics
• Sidon sets

Asks what can be said about the sequence starting $1,2$ in which each later term is the least larger integer that is uniquely a sum of two earlier terms, in particular about its density and its gaps.

• Number theory

Asks whether infinitely many $k$ make the completeness threshold of the $k$th powers larger than that of the $(k+1)$th powers, $T(n^k)>T(n^{k+1})$.

• Number theory
• Complete sequences

Asks for which $t,\alpha>0$ the sequence $\lfloor t\alpha^n\rfloor$ is complete, meaning that sums of distinct terms give all large integers.

• Number theory
• Complete sequences

Asks whether some $c>0$ makes every measurable set $A\subseteq\mathbb{R}^2$ of measure at least $c$ contain the vertices of a triangle of area $1$.

• Geometry

Asks whether $\lfloor 2^k\alpha\rfloor$ and $\lfloor 2^k\beta\rfloor$, for $\alpha,\beta>0$ with $\alpha/\beta$ irrational, together form a complete sequence, and the same for a base $\gamma\in(1,2)$. The first question is answered yes and the second has two readings.

• Number theory
• Complete sequences

Asks how fast the largest number $f(n)$ of integers up to $n$ with all sums of consecutive terms distinct grows, and whether $f(n)=o(n)$.

• Number theory

Asks what can be said about the density of the sequence starting at $n$ in which each later term is the least integer that is not a sum of consecutive earlier terms.

• Number theory

Asks for the size of the largest $A\subseteq\{1,\ldots,\lfloor cn\rfloor\}$ with no subset summing to $n$, and whether this size depends on $n$ in an irregular way.

• Number theory

Asks whether three consecutive positive integers can all be powerful, meaning that every prime dividing such a number divides it at least twice.

• Number theory
• Powerful numbers

Asks whether one of $n$ and $n+1$ must be a square when both are powerful, and whether the number of such $n\le x$ is at most $(\log x)^{O(1)}$.

• Number theory
• Powerful numbers

Asks whether some integer divisible by the square of each of its prime factors is followed by one divisible by the cube of each of its prime factors. Erdős and Graham's companion question, with the cube first, is answered by 12167, 12168.

• Number theory
• Powerful numbers

Asks whether, for every fixed $k$, the product of the powerful parts of $k$ consecutive integers near $n$ is at most about $n^2$.

• Number theory
• Powerful numbers

Asks how large the largest prime factor of $n(n+1)$ is.

• Number theory

Asks whether $n!=a_1!a_2!\cdots a_k!$ with $n-1>a_1\ge a_2\ge\cdots\ge a_k\ge2$ has only finitely many solutions.

• Number theory
• Factorials