Open-problem statements, with their sources attached.

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

Let $p_k$ be the $k$-th prime number. Are there infinitely many $n$ such that $(p_n + p_{n+2}) / 2$ is prime?

Let $p_k$ be the $k$-th prime number. Are there infinitely many $n$ such that $p_n = \dfrac{\sum_{i = 1} ^ k p_{n - i} + p_{n + i}}{2*k}$?

The Bateman-Horn Conjecture* Given a finite collection of distinct irreducible polynomials non-constant $f_1, f_2, \dots, f_k \in \mathbb{Z}[x]$ with positive leading coefficients that satisfy the Schinzel condition, the number of positive integers n ≤ x for which all polynomials $f_i$ are simultaneously prime is asymptotic to: $C(f_1, f_2, \dots, f_k) x / (log x)^k$ where $C$ is the Bateman-Horn constant given by the convergent infinite product: $C = \frac{1}{D}\prod_{p\in\mathbb{P}} (1 - 1/p)^(-k) · (1 - \omega_p/p)$ Here $\omega_p/p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes.

The Schinzel condition ensures that for each prime $p$, there exists some integer $n$ such that $p$ does not divide the product $f_(n) f_2(n) \dotsb f_(n)$, which guarantees the infinite product converges to a positive value.

The Beal Conjecture: if we are given positive integers $A, B, C, x, y, z$ such that $x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor.

Same parity betrothed numbers conjecture.* Do there exist betrothed numbers $(m, n)$ where both have the same parity (both even or both odd)?

All known betrothed pairs consist of one even and one odd number.

Infinitude of betrothed numbers conjecture.* Are there infinitely many betrothed number pairs?

Brocard's Conjecture* For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.

Büchi's problem* There exists a positive integer $M$ such that, for all integers $x$ and $a$, if $(x+n)^2 + a$ is a square for $M$ consecutive values of $n$, then $a = 0$.

*Büchi's problem (first open case, $M = 5$)**: For all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$, then $a = 0$.

Non-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case.

Bunyakovsky conjecture* If a polynomial $f$ over integers satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers $m$ such that $f(m)$ is prime.

Carmichael's totient function conjecture*: For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$.

For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation $ax^n - by^m = c$ where $(m, n) \neq (2, 2)$ and $x, y > 1$.

Lebesgue-Nagell Equation Conjecture*

For any odd prime $p$, the only integer solutions $(x, y)$ to the equation $x^2 - 2 = y^p$ are $(x, y) = (\pm 1, -1)$. Reference:* Ethan Katz and Kyle Pratt, "On the Lebesgue-Nagell equation $x^2 - 2 = y^p$", arXiv:2507.12397

There are infinitely many real quadratic fields ℚ(√d) with class number one, where d > 1 is a squarefree integer.

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 Collatz 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.

Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.