Asks whether every sufficiently large integer has the form $ap^2+b$ with $p$ prime, $a\ge1$ and $0\le b<p$.

• Number theory

Asks whether two disjoint blocks of $k$ consecutive integers can have the same least common multiple. Erdős's 1979 conjecture that they cannot is open, and the few known solutions all use blocks of different lengths.

• Number theory

Asks whether, for fixed $\epsilon>0$, infinitely many $n$ have $\omega(n-k)<(1+\epsilon)\log k/\log\log k$ for all large $k<n$, where $\omega$ counts distinct prime factors.

• Number theory

Asks whether every sufficiently large $n$ has some $k$ with $p(n+k)>k^2+1$, where $p(m)$ is the least prime factor of $m$.

• Number theory
• Primes

Asks whether every large $n$ has some $k$ such that $n+k$ is composite and $p(n+k)>k^2$, where $p(m)$ is the least prime factor of $m$.

• Number theory
• Primes

Asks whether, for some constant $c>0$, the largest prime divisor of $\binom{n}{k}$ is at least $\min(n-k+1,k^{1+c})$ for all $1\le k\le n$.

• Number theory
• Primes
• Binomial coefficients

Writes $\binom{n}{k}=uv$, where the prime factors of $u$ are at most $k$ and those of $v$ lie in $(k,n]$, and asks for bounds on the least $k$ with $u>n^2$.

• Number theory
• Primes
• Binomial coefficients

Asks whether, for $n^\epsilon<k\le n^{1-\epsilon}$, the number of distinct prime divisors of $\binom{n}{k}$ is $(1+o(1))k\sum_{k<p<n}1/p$.

• Number theory
• Primes
• Binomial coefficients

Asks whether every integer $N\ge2$ is a ratio $\prod_{i=1}^k(m+i)/\prod_{i=1}^k(n+i)$ of products of $k$ consecutive integers, for some $k\ge2$ and $m\ge n+k$.

• Number theory

Asks for the order of the longest initial interval that one residue class for each prime $p\le x$ can cover, the covering form of Jacobsthal's function. It is open: a pending partial claim gives $Y(x)\ll x^2/(\log x)^2$, so $Y(x)=o(x^2)$, and the order lies between $x\log x$ times iterated logarithms and that bound.

• Number theory

Asks for the largest $\epsilon_n$ such that one residue class for each prime $p$ with $n^{\epsilon_n}<p\le n$ covers every integer in $[1,n]$, and whether $\epsilon_n\to0$. It is open, between Erdős's lower bound and a counting bound of $1/e$.

• Number theory

Asks for a necessary and sufficient condition on a set of positive integers for its set of multiples to have density $1$; open, with one family of block sequences settled by Tenenbaum.

• Number theory

Asks for an estimate of the largest gap between consecutive integers in $[n,n^k]$, for fixed $k\ge2$, that have a divisor in $(n,2n)$, and whether it is at most $(\log n)^{O(1)}$.

• Number theory
• Divisors

Asks whether primes $p_1<p_2<\cdots$ with $p_{i+1}\equiv1\pmod{p_i}$ must have $p_k^{1/k}\to\infty$, and whether such a sequence exists with $p_k\le\exp(k(\log k)^{1+o(1)})$.

• Number theory

Asks whether, for every $1\le i<j\le n/2$, some prime $p\ge i$ divides both $\binom{n}{i}$ and $\binom{n}{j}$.

• Number theory
• Binomial coefficients

Asks, for composite $n$ and $f(n)=\min_{1<k\le n/2}\gcd(n,\binom{n}{k})$, which $n$ have $f(n)=n/P(n)$, where $P(n)$ is the largest prime factor of $n$, whether $f(n)>n^{1/2}$ infinitely often, and whether $f(n)\ll_A n/(\log n)^A$ for every $A>0$.

• Number theory
• Binomial coefficients

Asks to estimate the chromatic number of the unit distance graph in $\mathbb{R}^n$, whose edges join points at distance exactly $1$.

• Graph theory
• Geometry
• Chromatic number

Asks for an estimate of $L(r)$, the largest chromatic number of a graph on finitely many points in $\mathbb{R}^2$ whose edges join pairs at one of $r$ prescribed distances, and whether $L(r)\le r^{O(1)}$.

• Graph theory
• Chromatic number

Asks for bounds on the fewest integers from a run of $\max(A)$ consecutive integers whose product is divisible by $\prod_{a\in A}a$, for any set $A$ of $n$ integers. The corrected at-most statement is open, with a partial bound of $12n$ pending.

• Number theory

Asks for good bounds on, or an asymptotic formula for, the least $f(n)$ such that any $f(n)\max(A)$ consecutive integers hold distinct multiples of the $n$ elements of $A$, for every $n$-element set $A$. Known bounds are $\log n/\log\log n\ll f(n)\ll\sqrt n$.

• Number theory

Asks to prove $\max_mf(n,m)\le n^{1+o(1)}$ and $\max_m(f(n,m)-f(n,n))\to\infty$, where $(m,m+f(n,m))$ is the shortest interval holding distinct $a_1,\ldots,a_n$ with $k\mid a_k$. The second was proved by van Doorn (2026); the first is open, with an upper bound of order $n^{3/2}$ proved.

• Number theory

Asks for the Turán density of the complete $r$-uniform hypergraph on $k$ vertices, for any $k>r>2$. Turán's 1941 theorem settles $r=2$, no pair with $r>2$ is known, and Erdős's two prize offers stand.

• Graph theory
• Turán numbers
• Hypergraphs

Asks whether every bipartite graph $G$ with at least two edges has $\mathrm{ex}(n;G)\sim cn^\alpha$ for some $c>0$ and $\alpha\in[1,2)$, and whether $\alpha$ must be rational. The exponent forms Erdős proposed in 1967 were disproved in 1970.

• Graph theory
• Turán numbers

Asks whether the maximum number of edges in a $K_{r,r}$-free graph on $n$ vertices is $\gg n^{2-1/r}$.

• Graph theory
• Turán numbers

Asks whether every $r$-uniform hypergraph on $n$ vertices is a union of at most $\mathrm{ex}_r(n;K_{r+1}^r)$ edges and $(r+1)$-cliques, no two sharing an edge. It is the Erdős–Sauer conjecture, known for $r=2$.

• Graph theory
• Hypergraphs