Open-problem statements, with their sources attached.

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.

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