Asks whether there is a set $A\subseteq\mathbb{N}$ with $\liminf\lvert A\cap[1,x]\rvert/\pi(x)>0$ such that, for infinitely many $n$, $n-a$ is prime for every $a\in A$ with $0<a<n$.

• Number theory
• Primes

Asks whether, for all large $n$, the decreasing sequence with $a_1=n-1$ and each $a_k$ the greatest integer in $[2,a_{k-1})$ whose prime factors all exceed $n-a_k$ must contain a composite term.

• Number theory

Asks how dense $A+B$ can be for infinite sets $A,B\subseteq\mathbb{N}$ when the elements of $A+B$ are pairwise coprime.

• Number theory

Asks whether $\Lambda(k,2)$ is finite for all $k$ and $\Lambda(k,3)$ for all odd $k$, and how large they are, where $\Lambda(k,m)$ is the limsup over primes $p$ of the least start of $m$ consecutive $k$th power residues modulo $p$.

• Number theory

Asks whether, for every $c>1/2$ and every large enough prime $p$, each interval $(n,n+p^c)$ contains $a,b$ with $ab\equiv1\pmod p$.

• Number theory

Asks how large $y=y(\epsilon,n)$ must be for at most $\epsilon y$ of the integers in $(x,x+y)$ to have a divisor in $(n,2n)$.

• Number theory
• Divisors

Asks for estimates of $n_k$, the least integer $>2k$ such that $\prod_{1\le i\le k}(n_k-i)$ has no prime factor in $(k,2k)$. A 2026 preprint claims that $n_k$ grows faster than any power of $k$, and an elementary bound makes it at most exponential in $k$.

• Number theory

Asks for the size of the largest interval $I\subseteq[x,2x]$ on which every $n$ has more than $\log\log n$ distinct prime factors.

• Number theory

Asks whether $f(n)=\min_{i<n}(p_{n+i}+p_{n-i})$, the least sum of two primes placed symmetrically around $p_n$, satisfies $\limsup_n(f(n)-2p_n)=\infty$.

• Number theory
• Primes

Asks whether every sequence of primes $q_1<q_2<\cdots$ with non-decreasing gaps must satisfy $q_n/n^2\to\infty$.

• Number theory

Asks whether $[1,\ldots,p_{k+1}-1]<p_k[1,\ldots,p_k]$ for all $k\ge1$, where $[1,\ldots,n]$ is the least common multiple of $1,\ldots,n$.

• Number theory
• Primes

Asks whether $\sum_{0<a_i<n}1/a_i\to\infty$ for the greedy sequence with $a_0=0$, $a_1=1$ and each $a_k$ the least integer above $a_{k-1}$ with $n-a_k$ coprime to every earlier $n-a_i$, and the same for two complementary parts of the sum.

• Number theory

Asks whether, uniformly for all $t$ and $n$, the $t$-smooth parts of the $t$ integers in $[n+1,n+t]$ take $\gg t$ distinct values, where the $t$-smooth part of $m$ is the product of its prime factors below $t$, counted with multiplicity.

• Number theory
• Primes

Asks whether there is a constant $C>0$ with $\sum_{x\le n\le x+Cx^{1/2}(\log x)^2}p(n)/n\gg1$ for all large $x$, where $p(n)$ is the least prime factor of $n$.

• Number theory
• Primes

Asks whether there is a function $f$ with $f(n)\to\infty$ such that, for all large $n$, some composite $m$ satisfies $n+f(n)<m<n+p(m)$, where $p(m)$ is the least prime factor of $m$.

• Number theory
• Primes

Asks whether, for all large $x$, one can choose a residue class for each prime $p\le x$ and split these primes into two non-empty sets so that every $n<x$ lies in the chosen class of some prime in each set; open, with one sentence of the 1980 monograph as its only source.

• Number theory

Concerns the set $D_n$ of partial sums $d_1,d_1+d_2,\ldots$ of the divisors $1<d_1<d_2<\cdots$ of $n$. It asks how many elements of $D_n$ lie in no earlier $D_m$, and whether the least $n$ with $N\in D_n$ is $o(N)$.

• Number theory
• Divisors

Asks whether there are odd weird numbers and infinitely many primitive ones, where $n$ is weird if $\sigma(n)\ge2n$ but $n$ is not a sum of distinct proper divisors of $n$, and primitive if no proper divisor of $n$ is weird.

• Number theory
• Divisors

Asks whether some finite starting sequence of primes $q_1<\cdots<q_m$ makes infinite the sequence in which each next term $q_{n+1}$ is the least prime of the form $q_n+q_i-1$.

• Number theory

Asks whether every finite set of nonzero residues modulo a prime can be ordered with all partial sums distinct (Graham's rearrangement conjecture); four range results prove it for all large primes with no explicit threshold.

• Number theory
• Additive combinatorics

Asks whether, for a prime $p$, the set $A_p$ of residues $k!\bmod p$ with $1\le k<p$ has $|A_p|\sim(1-1/e)p$.

• Number theory
• Factorials

Asks whether, for every $k\neq1$, there are infinitely many $n$ with $2^n\equiv k\pmod n$.

• Number theory

Asks for an estimate of the least $N=f(k)$ such that every $k$-coloring of $\{1,\ldots,N\}$ has a monochromatic solution of $a+b=c$, and whether $f(k)$ is at most exponential in $k$. It is open, with $f(k)$ between a constant times $3.28^k$ and $(e-\frac16)k!$.

• Number theory
• Additive combinatorics
• Ramsey theory

Asks whether $\liminf_{n\to\infty} n\|n\alpha\|\|n\beta\|=0$ for all real $\alpha,\beta$, where $\|x\|$ is the distance from $x$ to the nearest integer.

• Diophantine approximation
• Number theory

Asks for $\mathrm{ex}_3(n,K_4^3)$, the largest number of triples on $n$ vertices with no four vertices carrying all four of their triples (Turán's tetrahedron problem). The density lies between Turán's $5/9$ and a flag-algebra bound of $0.5615$.

• Graph theory
• Hypergraphs
• Turán numbers