Open-problem statements, with their sources attached.

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.

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

Dickson's conjecture* If a finite set of linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition, there exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all $i$.

Polignac's conjecture* For any integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.

The infinitude of Sophie Germain primes* There are infinitely many primes $p$ such that $2p + 1$ is prime.

The infinitude of cousin primes* There are infinitely many primes $p$ such that $p + 4$ is prime.

The infinitude of sexy primes* There are infinitely many primes $p$ such that $p + 6$ is prime.

The Elliott–Halberstam conjecture: for every $\theta < 1$ and $A > 0$ there exists a constant $C > 0$ such that $\sum_{1 \le q \le x^{\theta}} E(x; q) \le \frac{C x}{\log^A x}$ for all $x > 2$.