Open-problem statements, with their sources attached.

Schinzel conjecture (H hypothesis)* If a finite set of polynomials $f_i$ satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers $n$ such that $f_i(n)$ are primes for all $i$.

The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer $n$, the addition-chain length of $2^n - 1$ is at most $n - 1 + \ell(n)$.

PSW conjecture* (Selfridge's test) Let $p$ be an odd number, with $p \equiv \pm 2 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p+1} \equiv 0 \pmod{p}$, then $p$ is a prime number.

Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow monotonically in $n$.

The Sierpiński problem (Selfridge's conjecture).* Is 78557 the smallest Sierpiński number?

Selfridge conjectured that 78557 is the smallest Sierpiński number. He proved in 1962 that 78557 is indeed a Sierpiński number by showing that all numbers of the form $78557 \cdot 2^n + 1$ have a factor in the covering set $\{3, 5, 7, 13, 19, 37, 73\}$.

The prime Sierpiński problem.* Is 271129 the smallest prime Sierpiński number?

In 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime $k = 271129$.

The extended Sierpiński problem.* Is 271129 the second-smallest Sierpiński number?

Even if 78557 is confirmed as the smallest Sierpiński number, there could exist a composite Sierpiński number $k$ with $78557 < k < 271129$. We formalize "second-smallest" as: the least Sierpiński number $k$ such that there exists exactly one Sierpiński number below it.

Singmaster's conjecture: the number of times any number $t > 1$ appears in Pascal's triangle is bounded.

Is 10 a solitary number?* The smallest positive integer whose solitary status is currently unresolved is $10$, with abundancy index $\sigma(10) / 10 = 9/5$.

Existence of an infinite club.* A club is an abundancy equivalence class, i.e. the set of all positive integers friendly with a given $n$. It is unknown whether any club is infinite.

An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9.

There does not exist a $(2,5)$-perfect number

Taxicab number for $k=5$, $m=2$, and $n=2$ is not known. Whether such a number exists is also not known.

Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$. Whether such a number exists is also not known.

$e + \pi$ is transcendental.

$e\pi$ is transcendental.

$e^{\pi^2}$ is transcendental.

$e^e$ is transcendental.

$\pi^e$ is transcendental.

$\pi^{\sqrt{2}}$ is transcendental.