Open-problem statements, with their sources attached.

*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.

All members of the sequence satisfy $n \equiv 108 \pmod{216}$.

$108$ is the only primitive term.

For members of the sequence other than $8$, we have $k + 1$ is prime.

*Conjecture (A81091)**: There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$.

Let $T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}$. If $T_N$ is even (equivalently $N \equiv 0 \pmod 4$ or $N \equiv 3 \pmod 4$), then under optimal play the game Catch-Up($\{1, \ldots, N\}$) ends in a draw.

Every even number greater than 4208 is the sum of two twin primes.

Kurepa's conjecture

For all $n$, $!n\not\equiv 0 \mod n$

This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy