Open-problem statements, with their sources attached.

It seems very likely that we must have $y \ll x^{1+o(1)}$ [Gr24].

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?

The following very particular instance is probably the simplest [Gr24]: Suppose that $A \subset \mathbb{N}$ is a set with the property that $|A \pmod p| \leqslant \frac{1}{2}(p + 1)$ for all sufficiently large $p$. Is it true that either $|A \cap [X]| \ll X^{1/2} / \log^{100} X$, or $A$ is contained in the image of $\mathbb{Z}$ under a quadratic map $\phi : \mathbb{Q} \to \mathbb{Q}$?

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 $|A + A| ≥ |A|^{1 + c}$?

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 do exist such $A$ with $|A| \ll_C n / \log^C n$ for any $C$.

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 $a_1, a_2 \in A$?

Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? We formalize this as an eventual statement for sufficiently large real $X$.

Does Ulam's sequence have positive density?

1. implies 2): if the coefficients of $f$ satisfy the $\omega$-integrality condition for some superlinear $\omega$, then there exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$.

1. implies 1): if the coefficients of $f$ are in $\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\mathbb{Q}[z]$. Also the version of conjecture of Litt's problem 1 on his website.

If $2^x$ and $3^x$ are integers, then $x$ must be an integer.

Is 2n the complexity of 2^n for 0 < n?

The Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part $\frac{1}{2}$. That is, if $\zeta(s) = 0$, $s \neq 1$, and $s$ is not a trivial zero $-2(n+1)$ for some $n \in \mathbb{N}$, then $\operatorname{Re}(s) = \frac{1}{2}$.

This is the official Millennium Prize Problem as posed by the Clay Mathematics Institute.

This uses the RiemannHypothesis type from Mathlib, which is defined as ∀ (s : ℂ), riemannZeta s = 0 → (¬∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2.

The Generalized Riemann Hypothesis asserts that all the non-trivial zeros of the Dirichlet $L$-function $L(\chi, s)$ of a primitive Dirichlet character $\chi$ have real part $\frac{1}{2}$.

Conjecture: the sequence A228828 is infinite.

The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime.

*Zhi-Wei Sun's Conjecture (A232174)**: Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

*Zhi-Wei Sun's Conjecture (A239957)**: Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer.

*Zhi-Wei Sun's 1680-Conjecture (A280831)**: Any nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.

*Zhi-Wei Sun's Conjecture (A281976)**: Any integer $n \geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers and $z \leq w$, such that both $x$ and $x + 24y$ are squares.

*Zhi-Wei Sun's Conjecture (A287616)**: Any nonnegative integer can be written as the sum of a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized heptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.