Open-problem statements, with their sources attached.

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.

*Zhi-Wei Sun's Conjecture (A303656)**: Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5.

*Zhi-Wei Sun's 2-4-6-8 Conjecture (A306477)**: Any integer $n > 0$ can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.

*Zhi-Wei Sun's Four-Square Conjecture (A308734)**: Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime.

A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime.

Conjecture: $a(n) = O(\log(n)\log(\log(n)))$.

Counter-conjecture to a_isBigO: $a(n) / (\log n \log \log n)$ is unbounded.

We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$, with a finite number of exceptions that depend on $m$.

There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$. See comment by Zhi-Wei Sun (Dec 02 2013).

All members of the sequence A56777 come from prime quadruples.