Open-problem statements, with their sources attached.

Let $P = (m_1, \dots, m_k)$ be a tuple of positive even integers. Let $\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$ forms an admissible prime constellation. Let $w(q; m_1, \dots, m_k)$ denote the number of distinct residues of $0, m_1, \dots, m_k$ modulo $q$, and let

Then

For integers $x, y \geq 2$,

where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$.

Idoneal numbers completeness conjecture.

Is the Catalan constant $G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$ irrational?

Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The Juggler Conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is $1$.

Kummer–Vandiver conjecture states that for every prime $p$, the class number of the maximal real subfield of $\mathbb{Q}(\zeta_p)$ is not divisible by $p$.

The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers equals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from all values on the right, then $n + m \geq k$.

Formally, for positive integers $k, n, m \in \mathbb{N}$ and sequences $x : \{0, \ldots, n-1\} \to \mathbb{N}$ and $y : \{0, \ldots, m-1\} \to \mathbb{N}$ with $x_i > 0$, $y_j > 0$, and $x_i \neq y_j$ for all $i, j$, if $\sum_{i=0}^{n-1} x_i^k = \sum_{j=0}^{m-1} y_j^k,$ then $k \leq n + m$.

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