Open-problem statements, with their sources attached.

Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$, where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n) = \infty$?

Is it true that, for every prime $p$, there is a prime $q \leq p$ which is a primitive root modulo $p$?

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

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\frac{1}{10} N$?

We interpret "half the residue classes" as $\lfloor p_i / 2 \rfloor$.

We conjecture that the best-known lower bound can be improved.

We conjecture that the best-known upper bound can be improved.

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 $\frac{1}{2}$.