Open-problem statements, with their sources attached.

Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive integers to $x_1^5 + x_2^5 + x_3^5 = y^5.$ That is, for all $x_1, x_2, x_3, y \in \mathbb{N}$ with $x_1, x_2, x_3, y > 0$, $x_1^5 + x_2^5 + x_3^5 \neq y^5.$ This corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture, where $n + m = 4 < 5 = k$ would be required to yield a counterexample.

Does there always exist at least one prime between consecutive perfect squares?

Let M(f) denote the Mahler measure of f. There exists a constant μ>1 such that for any f(x)∈ℤ[x], M(f)>1 → M(f)≥μ.

μ=M(X^10 + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1) is the best value for lehmer_mahler_measure_problem.

Does there exist a composite number $n > 1$ such that Euler’s totient function $\varphi(n)$ divides $n - 1$?

For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.

For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$

For any two real numbers $\alpha$ and $\beta$,

where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the nearest integer.

For real number $\alpha$ and prime $p$,

where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the nearest integer, and $|x|_{p}$ is the $p$-adic norm.

Consider $n$ runners on a circular track of unit length. At the initial time $t = 0$, all runners are at the same position and start to run; the runners' speeds are constant, all distinct, and may be negative. A runner is said to be lonely at time $t$ if they are at a distance (measured along the circle) of at least $\frac 1 n$ from every other runner. The lonely runner conjecture states that each runner is lonely at some time, no matter the choice of speeds.

Lychrel conjecture (base 10):* conjecturally, there are no Lychrel numbers in base 10.

Equivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.

The first widely studied open case: whether 196 is a base-10 Lychrel number.

Does there exist a $3 \times 3$ matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value?

0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn. See Magic Square of Squares

The Mahler Conjecture states that there are no non-zero Z-numbers.

For any odd natural number p if two of the following conditions hold, then all three must hold:

1. $2^p-1$ is prime
2. $(2^p+1)/3$ is prime
3. Exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$

The New Mersenne Conjecture statement holds for odd primes.

Are there infinitely many Mersenne primes?

The first five Catalan-Mersenne numbers $c_0, \ldots, c_4$ are known to be prime. Catalan conjectured that they are prime "up to a certain limit". Are all Catalan-Mersenne numbers $c_n$ with $n \geq 5$ prime?

$\pi$ is normal in base 10.

For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$.