Open-problem statements, with their sources attached.

The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental.

There are no distinct primes $p$ and $q$ such that $\frac{q^p - 1}{q - 1}$ divides $\frac{p^q - 1}{p - 1}$

Are Fermat numbers composite for all n > 4?

Are there infinitely many Fermat primes?

Are there infinitely many composite Fermat numbers?

Are all Fermat numbers are square-free?

The Fermat–Catalan conjecture states that the equation $a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values $(a^m, b^n, c^k)$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying $\frac 1 m + \frac 1 n + \frac 1 k < 1$.

There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime It is also a barrier to defining a benchmark from this paper: https://arxiv.org/html/2505.13938v1 (see Figure 8).

There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime.

Firoozbakht's conjecture* The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$.

It is conjectured that the correct bound is

[Ha59] Hardy, G. H. (1959). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work(3rd ed.). New York: Chelsea Publishing Company. p. 67

See also https://arxiv.org/abs/2305.03549

Gilbreath's conjecture* Gilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1.

Can every even integer greater than 2 be written as the sum of two primes?

Grimm's Conjecture* If $n, n+1, \dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$ such that $p_i$ divides $n + i$ for all $0 \le i \le k-1$.

Grimm's Conjecture, weaker version* If $n, n+1, \dots, n+k-1$ are all composite numbers, then their product has at least $k$ distinct prime divisors.

Original Hall's conjecture with exponent $1/2$.

Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \varepsilon$.

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.