Browse Problems

Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $n$, then $$\frac{1}{\Phi_n(p\boxplus_n q)} \ge \frac{1}{\Phi_n(p)}+\frac{1}{\Phi_n(q)}?$$ Note: while no proof of this is published yet, the authors of …

Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $2$, then $$\frac{1}{\Phi_2(p\boxplus_n q)} \ge \frac{1}{\Phi_2(p)}+\frac{1}{\Phi_2(q)}?$$

Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $3$, then $$\frac{1}{\Phi_3(p\boxplus_n q)} \ge \frac{1}{\Phi_3(p)}+\frac{1}{\Phi_3(q)}?$$

Is there an Euler brick in $4$-dimensional space?

**Four exponentials conjecture** Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental.

`F n` is the smallest number of vertices of a triangle-free graph with chromatic number `n` and `f = 3`.

Conjecture: The gcd condition is equivalent to the prime power condition.

**Gilbreath's conjecture** Gilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1.

Can every even integer greater than 2 be written as the sum of two primes?

The Gourevitch series identity: The following idenitity holds: $\sum_{n=0}^{\infty} \frac{1 + 14 n + 76 n^2 + 168 n^3}{2^{20 n}} \binom{2n}{n}^7 = \frac{32}{\pi^3}.$ This was originally conjectured in [G2003] by Guillera and …

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in …

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?

The answer is YES for 4-term progressions [BJP14].

What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?

$f(N) \gg N \cdot e^{-c(\log N)^{1/7}}$.

From [Ruzsa] $f(N) \gg N^{1/2}$.

From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$.

Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ …

[So13] proved this is true for "BMZ corners". Follows from the proof of Theorem 2.1, p.1456-1457.

What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? Suppose that $A \subset \mathbb{F}_2^n \times \mathbb{F}_2^n$ is a set …

[Ma21] showed that $3.13 \leq C$.

[Ma21] showed that $C \leq 4$.

Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size $(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?

Suppose that $\mathbb{N}$ is finitely coloured. Are there $x,y$ of the same colour such that $x^2 + y^2$ is a square? Solved in [FrKlMo25].

If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\lbrace 0,1,3 \rbrace$ that $A$ can contain? Conjectured in [Aa19] p.579: $\left({1}{3} + …

From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$.

From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$.

From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$.

From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$.

Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?

Lower bound for $c(p)$ for $1 < p \le \infty$, improving the known value at $p = 2$ or $p = \infty$.

Upper bound for $c(p)$ for $1 < p \le \infty$, improving the best-known value at $p = \infty$.

Given a natural number `N`, what is the smallest size of a subset of `ℕ` that contains, for each `d = 1, …, N`, an arithmetic progression of length `k` with common …

Asymptotic version: determine the asymptotic behavior of `m(N, k)` as `N` grows. The solver should determine what function `f : ℕ → ℝ` eventually equals `(fun N ↦ (m N k : …

Determine an upper bound (big O) for `m(N, k)`.

Determine a strict upper bound (little o) for `m(N, k)`.

Determine the asymptotic equivalence class (theta) of `m(N, k)`.

What is the largest product-free set in the alternating group $A_n$?

Is it true that for every finite abelian group $G$ the convex hulls $\Phi(G)$ and $\Phi'(G)$, obtained from kernels $\phi(g) = \mathbb{E}_{x_1 + x_2 + x_3 = g} f_1(x_2, x_3) f_2(x_1, x_3) …

Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \mathbb{N}$ is a set of squares with $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| …

Suppose that $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1 - o(1)}$? It is known that necessarily $|A| \geq n^{2/3 - o(1)}$, whilst in the other direction there …

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \ldots, p-1\}$ congruent to some product $a_1 a_2$ where …

**Green's Open Problem 72 / No-three-in-line problem**: The no-k-in-line conjecture holds for $k = 3$.

Does the no-three-in-line problem hold when $N$ is big enough?

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?