Open-problem statements, with their sources attached.

*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.

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.