Browse Problems

Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where $p = \lfloor\frac d2\rfloor$.

Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$ such that $\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \le \frac{p - 1}{2p} …

Any $A\subseteq \mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$ are infinite. The Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19].

Are there infinitely many binomial coefficients with deficiency 1?

Are there only finitely many binomial coefficients with deficiency > 1?

For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$, with only finitely many exceptions.

[Sorenson, Sorenson, and Webster](https://mathscinet.ams.org/mathscinet/relay-station?mr=4235124) give heuristic evidence that \[\log g(k) \asymp \frac{k}{\log k}.\]

Erdős, Lacampagne, and Selfridge [ELS93](https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199990-6/S0025-5718-1993-1199990-6.pdf) write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

The current record is\[g(k) \gg \exp(c(\log k)^2)\]for some $c>0$, due to Konyagin [Ko99b](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300007555).

Ecklund, Erdős, and Selfridge conjectured $g(k)\leq \exp((1+o(1))k)$ [EES74](https://mathscinet.ams.org/mathscinet/relay-station?mr=1199990)

Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

The current record is $g(k) \gg \exp(c(\log k)^2)$ for some $c>0$, due to Konyagin [Ko99b]. -

Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\leq \exp((1+o(1))k)$.

The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$.

Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers …

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. Ref: …

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there …

Bogdan Grechuk has observed that `1117175146` is not the sum of a prime and at most $3$ powers of $2$.

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there …

There are infinitely many even integers not the sum of a prime and $2$ powers of $2$

Is every odd $n > 1$ the sum of a squarefree number and a power of 2?

1. There is NO good sequence with polynomial growth.

2. There is a good sequence with sub-exponential growth.

If `A = {a₁ < a₂ < …}` has property P, then `A` has natural density `0`. Equivalently, `(a_j / j) → ∞` as `j → ∞`.

Conversely, for any function `f : ℕ → ℕ` that goes to infinity, there exists a strictly increasing sequence `A = {a₁ < a₂ < …}` with property P such that `(a_j …

There exists an infinite sequence $A = {a₁ < a₂ < …} ⊂ \mathsf{SF}$ where $\mathsf{SF} := \mathbb{N} \setminus \bigcup_{p} p^{2}\mathbb{N}$, i.e. the set of squarefree numbers. The set `A` has property …

Every sequence with property Q has upper density at most `6 / π^2`.

Lower bound (Hefty–Horn–King–Pfender 2025). There exists a constant $c_1 \in (0,1]$ such that, for sufficiently large $n$, $$ c_1 \sqrt{\frac{n}{\log n}} \le f(n), $$ where $f(n)$ denotes the maximum chromatic number of …

Upper bound (Davies–Illingworth 2022). There exists a constant $c_2 \ge 2$ such that, for sufficiently large $n$, $$ f(n) \le c_2 \sqrt{\frac{n}{\log n}}, $$ where $f(n)$ denotes the maximum chromatic number of …

Lower bound (Hefty–Horn–King–Pfender 2025). There exists a constant $c_1 \in (0,1]$ such that, for sufficiently large $n$, $$ c_1 \sqrt{\frac{n}{\log n}} \le f(n), $$ where $f(n)$ denotes the maximum chromatic number of …

Upper bound (Davies–Illingworth 2022). There exists a constant $c_2 \ge 2$ such that, for sufficiently large $n$, $$ f(n) \le c_2 \sqrt{\frac{n}{\log n}}, $$ where $f(n)$ denotes the maximum chromatic number of …

Is $\mathrm{AR}(n, C_k) = \left(\frac{k-2}{2} + \frac{1}{k-1}\right)n + O(1)$?

The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all …

Let $P_k$ be the path on $k$ vertices and $\ell=\lfloor\frac{k-1}{2}\rfloor$. If $n\geq k\geq 5$ then is $\mathrm{AR}(n,P_k)$ equal to $\max\left(\binom{k-2}{2}+1, \binom{\ell-1}{2}+(\ell-1)(n-\ell+1)+\epsilon\right)$where $\epsilon=1$ if $k$ is odd and $\epsilon=2$ otherwise? A proof of …

For $n \geq k \geq 5$, let $\ell = \lfloor(k-1)/2\rfloor$ and $\epsilon = 1$ if $k$ odd, else $2$. Is $\mathrm{AR}(n, P_k) = \max\left(\binom{k-2}{2}+1,\; \binom{\ell-1}{2}+(\ell-1)(n-\ell+1)+\epsilon\right)$?

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to …

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.

Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?

Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers.

For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?

Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?

Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that $$\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0$$ as $x\to \infty$?

Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that $$\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?$$

Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$? In [Va99] it is asked whether $968$ is the largest integer with this …

Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$. If $A+B$ contains all sufficiently large positive integers then is it …

Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\max(x^2,y^2,z^2)\leq n$?

[Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$.

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $$\max_{|z|=1} |P(z)| > (1 + c) …

Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the …