Open-problem statements, with their sources attached.

*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

This statement can be reduced to the prime case only.

An equivalent formulation in terms of the gcd of $n!$ and $!n$.

For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0. Suppose that for every prime p there exists an integer n such that p ∤ ∏ i, (aᵢ n + bᵢ). Then there exist infinitely many n such that aᵢ n + bᵢ is prime for all i.