AMO-Bench 2025 Problem 2

The solution set is the union of the following five sets: \begin{enumerate} \item $\{(2, n, n) | n\in\mathbb{N}^{+}\}$ \item $\{(2, 2n+3, 2n) | n\in\mathbb{N}\}$ \item $\{(2, 2n, 2n+3) | n\in\mathbb{N}\}$ \item $\{(2^{q}-1, kq+1, kq) | q\in\mathbb{N}^{+} \text{ such that } 2^{q}-1 \text{ is a prime (i.e., a ``Mersenne prime''), and } k\in\mathbb{N}\}$ \item $\{(2^{q}-1, kq, kq+1) | q\in\mathbb{N}^{+} \text{ such that } 2^{q}-1 \text{ is a prime (i.e., a ``Mersenne prime''), and } k\in\mathbb{N}\}$ \end{enumerate} Here, $\mathbb{N}^{+}$ denotes the set of positive integers, and $\mathbb{N}$ denotes the set of non-negative integers. Prime numbers of the form $2^{q}-1$ are also known as ``Mersenne primes''. (Note: For the cases where $p=2^{q}-1$, the sole requirement is that $2^{q}-1$ be a prime number; no further conditions are imposed on the exponent $q$.)