Browse Problems

Probably there is no such $A$ for the polynomial $X^3$.

Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?

Are there infinitely many integers $n, m$ such that $ϕ(n) = σ(m)$?

Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that $$\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447?$$ A conjecture of Newman.

This was proved by Chung and Graham \cite{ChGr84}, who in fact prove that $$\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq \frac{1}{c}\approx 0.3944$$ where $$c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.5353705\cdots$$ and $F_m$ is the $m$th Fibonacci number.

They also prove that this constant is best possible.

For each $n \in \mathbb{N}$ choose some $X_n \subseteq \mathbb{Z}/n\mathbb{Z}$. Let $B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}$. Must $B$ …

Let $A$ be a finite set and $$B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.$$ Is it true that, for every $m>n\geq \max(A)$, $$\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap …

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true …

When $A = \{p^2 : p \textrm{ prime}\}$, $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős. This is the $\alpha = 2$ case …

Similarly, Tao noted that the conjecture fails when $|A| = 2k$, by taking $A$ to be a set of the total sum 0 and considering $-A$.

Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when $|A|$ is sufficiently large.

Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and $|A| \ne 2^l$ for $l \ge 0$. They also gave counterexamples when $k = 2$ and $|A| …

Selfridge and Straus [SeSt58] also showed that the conjecture is true when 1) $k = 3$ and $|A| > 6$ or 2) $k = 4$ and $|A| > 12$. More generally, they …

Kruyt noted that the conjecture fails when $|A| = k$, by rotating $A$ around an appropriate point.

Let $\alpha,\beta \in \mathbb{R}$. Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]? This is also known as the Littlewood conjecture.

Let $M$ be a real $n \times n$ doubly stochastic matrix. Does there exist some $σ \in S_n$ such that $$ \prod_{1 \leq i \leq n} M_{i, σ(i)} \geq n^{-n}? $$ This …

A weaker version of Erdős' problem 499, which asks whether for every doubly stochastic matrix, there exists a permutation $σ \in S_n$ with $M_{i, σ(i)} ≠ 0$ and such that $$ \sum_{1 …

What is the size of the largest $A \subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x$, $y$, $z$ from $A$, …

Alweiss has observed a lower bound of $\binom{n + 1}{2}$ follows from considering the subset of $\mathbb{R}^{n + 1}$ formed of all vectors $e_i + e_j$ where $e_i$, $e_j$ are distinct coordinate …

When $n = 2$, the answer is 6 (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c]). [ErKe47] Erdős, Paul and Kelly, L. M., Elementary Problems and Solutions: …

When $n = 3$, the answer is 8 (due to Croft [Cr62]). [Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.

The best upper bound known in general is due to Blokhius [Bl84] who showed that $$ |A| \leq \binom{n + 2}{2} $$ [Bl84] Blokhuis, A., Few-distance sets. (1984), iv+70.

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.

Estimate a lower bound for$\alpha(n)$.

Estimate an upper bound for$\alpha(n)$.

Erdős observed that $\alpha(n) \gg 1/n^2$.

Current best lower bound [KPS82].

Current best upper bound [CPZ24]: $\alpha(n) \ll n^{-7/6 + o(1)}$.

It is trivial that $\alpha(n) \ll 1/n$.

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii …

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii …

Let $f(z) ∈ $ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii …

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. If it is connected, can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of circles the sum …

Is there an infinite set $A \subset \mathbb{N}$ such that for every $a \in A$, there is an integer n such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer, …

**Chowla's cosine problem** If $A\subset \mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some absolute constant $c>0$ and $\theta$ such that $$\sum_{n\in A}\cos(n\theta) < …

Bedert [Be25c] proved an upper bound of $-c N^{1/7}$.

Ruzsa [Ru04] proved an upper bound of $-\exp(O(\sqrt{\log N})$.

For all transcendental entire function `f`, `liminf (fun r : ℝ => ratio r f) atTop > 1 / 2`.

Let `f` be a transcendental entire function. What is the greatest possible value of `liminf (fun r : ℝ => ratio r f) atTop`?

For all transcendental entire function `f`, `liminf (fun r : ℝ => ratio r f) atTop ≤ 2 / π - c` for some `c > 0`. This is proved in [ClHa64].

Let `f = ∑ aₖzⁿₖ` be an entire function such that `nₖ > k (log k) ^ (2 + c)`. Then `limsup (fun r => ratio r f) atTop = 1`. This …

Let `f = ∑ aₖzⁿₖ` be an entire function of finite order such that `nₖ / k → ∞`. Then `limsup (fun r => ratio r f) atTop = 1`. This is …

Is it true that for all entire functions `f = ∑ aₖzⁿₖ` such that `∑' 1 / nₖ < ∞`, `limsup (fun r => ratio r f) atTop = 1`?

If `f(z) = ∑ aₖzⁿₖ` is an entire function (with `aₖ ≠ 0` for all `k`) such that `nₖ / k → ∞`, is it true that `f` assumes every value infinitely …