Asks whether any $k$ consecutive composite integers $n+1,\ldots,n+k$ can be assigned distinct primes $p_1,\ldots,p_k$ with $p_i\mid n+i$.

• Number theory
• Primes

Asks whether there are infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$.

• Number theory
• Binomial coefficients
• Base representations

Asks whether the sum of $1/p$ over the primes $p\le n$ that do not divide $\binom{2n}{n}$ is bounded by an absolute constant.

• Number theory
• Binomial coefficients

Asks whether an interval $[u,v]$ in which the largest prime dividing the product appears at least twice must have $v-u=v^{o(1)}$, and whether $v-u$ can be arbitrarily large.

• Number theory

Asks whether, for every $k$, infinitely many primes $p$ are the largest prime factor of $\prod_{0\le i\le k}(p^2+i)$.

• Number theory

Asks whether the maximum of $m+p(m)$ over composite $m<n$, where $p(m)$ is the least prime factor of $m$, exceeds $n$ for all large $n$, and whether the excess grows without bound.

• Number theory

Asks whether $\binom{n}{k}$ with $2\le k\le n-2$ can be a product of consecutive primes infinitely often, as $\binom{21}{2}=2\cdot3\cdot5\cdot7$ is.

• Number theory
• Binomial coefficients

Asks whether two disjoint blocks of more than three consecutive integers can have equal products only finitely often, and whether all such cases can be classified.

• Number theory

Asks whether every $n\ge1$ has some $k$ for which $n(n+1)\cdots(n+k-1)$ divides $(n+k)\cdots(n+2k-1)$.

• Number theory

Asks how $f(n)$ behaves, where $f(n)$ is the least $m\ge1$ such that $n!=a_1\cdots a_t$ with $a_1<\cdots<a_t=a_1+m$.

• Number theory
• Factorials

Asks whether, for every $k$, there is an $n$ with $n(n-1)\cdots(n-k)\mid\binom{2n}{n}$.

• Number theory
• Binomial coefficients

Asks, for each $k\ge2$, whether $\sum_{n\le x}g_k(n)\sim c_kx\log x$ and whether $g_k(n)=c_k\log x+o(\log x)$ for almost all $n<x$, where $g_k(n)$ is the largest value of $a_1+\cdots+a_k-n$ with $a_1!\cdots a_k!\mid n!$.

• Number theory
• Factorials

Asks for which $a\ge1$ and primes $p$ the exponents $k$ with $p^k\mid a_1!+\cdots+a_n!$ for some $a=a_1<\cdots<a_n$ are bounded, and how the largest such $k$ behaves.

• Number theory
• Factorials

Asks whether only finitely many powers of $2$ use only the digits $0$ and $1$ in base $3$.

• Number theory
• Base representations

Asks how many iterations of $n\mapsto\phi(n)+1$ are needed to reach a prime, and how often a given prime is reached.

• Number theory
• Iterated functions

Asks whether $\sigma_k(n)^{1/k}\to\infty$ as $k\to\infty$ for every $n\ge2$, where $\sigma_k$ is the $k$-th iterate of the sum-of-divisors function.

• Number theory
• Iterated functions

Asks for which $n$ and $r$ the iterates $g_k$ of $g(n)=n+\phi(n)$ satisfy $g_{k+r}(n)=2g_k(n)$ for all large $k$.

• Number theory
• Iterated functions

Asks whether, for all $m,n\ge2$, the iterated sum-of-divisors sequences starting at $m$ and at $n$ eventually share a value.

• Number theory
• Iterated functions

Asks whether infinitely many $n$ satisfy $m+\omega(m)\le n$ for every $m<n$, where $\omega(m)$ is the number of distinct prime factors of $m$.

• Number theory
• Iterated functions

Asks whether, under the map $n\mapsto n+\tau(n)$ with $\tau(n)$ the number of divisors, the iterates of any two starting integers eventually reach a common value.

• Number theory
• Iterated functions

Asks whether the ratio of the number of totient values up to $x$ to the number of distinct values $\phi(m)$ with $m\le x$ has a limit, and whether that limit exceeds 1; phase reductions conditional on an unrefereed manuscript leave convergence open, and an unrefereed non-arXiv note claims a conditional upper bound.

• Number theory

Asks whether $\tau((n+\lfloor(\log n)^C\rfloor)!)/\tau(n!)$ tends to infinity for large $C$, where $\tau$ counts divisors.

• Number theory

Asks how the sequence given by $f(1)=f(2)=1$ and $f(n)=f(n-f(n-1))+f(n-f(n-2))$ behaves, and whether it misses infinitely many integers.

• Number theory

Asks for the asymptotic behaviour of the sequence with $a_1=1$, $a_2=2$ in which each later term is the least integer above the previous one that is a sum of at least two consecutive earlier terms.

• Number theory

Asks for an estimate of the largest subset of $\{1,\ldots,n\}$ with distinct pairwise products, and a bound for sets whose products of $r$ increasing members are distinct.

• Number theory
• Sidon sets