Open-problem statements, with their sources attached.

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.

Zagier's conjecture*

The $\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values of weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence satisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$.

For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)

For every positive real number ε, there exists a constant K_ε such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have c < K_ε rad(abc)^(1+ε).

For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

The Agoh-Giuga Conjecture, Agoh's formulation

The Agoh-Giuga Conjecture, Giuga's formulation

Agrawal's Primality Conjecture.*

Does the congruence $(X-1)^n \equiv X^n - 1 \pmod{n, X^r-1}$ imply $n$ is prime (with a specific exception for $n^2 \equiv 1 \pmod{r}$)?

While the "if" direction is a known theorem, the "only if" direction remains a conjecture.

Roman B. Popovych Conjecture.* A stronger version of Agrawal's conjecture, which also considers the congruence $(X+2)^n \equiv X^n + 2 \pmod{n, X^r-1}$. If both congruences hold, then $n$ is either prime or $n^2 \equiv 1 \pmod{r}$. This variant was proposed by Roman B. Popovych in 2018.

Non-Power-of-2 Almost Perfect Numbers Conjecture.* Does there exist an almost perfect number that is not a power of 2?

Relatively prime amicable numbers conjecture.* Do there exist amicable numbers $(a, b)$ with $\gcd(a, b) = 1$?

All known amicable pairs share a common factor. It is an open question whether a pair of relatively prime amicable numbers can exist. Reference:* Wikipedia

Infinitely many amicable numbers conjecture.*

Are there infinitely many pairs of amicable numbers?

While many amicable pairs are known, it remains open whether there are infinitely many. Reference:* Wikipedia, erdosproblems.com/830

Amicable numbers with opposite parity conjecture.* Do there exist amicable numbers $(a, b)$ where one is even and the other is odd?

All known amicable pairs are either both even or both odd. It is widely believed that mixed-parity amicable pairs do not exist, but this remains open. Reference:* Wikipedia

Andrica's conjecture* The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.

Artin's Conjecture on Primitive Roots*, first half. Let $a$ be an integer that is not a square number and not $−1$. Then the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ has a positive asymptotic density inside the set of primes. In particular, $S(a)$ is infinite.

Artin's Conjecture on Primitive Roots*, second half. Write $a = a_0 b^2$ where $a_0$ is squarefree. Under the conditions that $a$ is not a perfect power and $a_0\not\equiv 1\pmod{4}$ (sequence A85397 in the OEIS), the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is independent of $a$ and equals Artin's constant.

Artin's Conjecture on Primitive Roots*, second half, power version If $a = b^m$ is a perfect odd power of a number $b$ whose squarefree part $b_0\not\equiv 1 \pmod{4}$, then the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is given by $C\prod_{p \mid m} \frac{p(p - 2)}{p^2 - p - 1}$, where $C$ is Artin's constant.

Artin's Conjecture on Primitive Roots*, second half, power version If $a = b^m$ is a perfect power of a number $b$ whose squarefree part $b_0\equiv 1 \pmod{4}$, then the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is given by