Question 114
For $n \in \mathbb{N}$, consider the system $(S_n) := \begin{cases} x^2 + n y^2 = z^2 \\ n x^2 + y^2 = t^2 \end{cases}$, where $x, y, z, t \in \mathbb{N}$. If $M_1 = \{ n \in \mathbb{N} \mid \text{system } (S_n) \text{ has infinitely many solutions} \}$, and $M_2 = \{ n \in \mathbb{N} \mid \text{system } (S_n) \text{ has no solutions} \}$, prove that:
a) $7 \in M_1, 10 \in M_2$;
b) sets $M_1$ and $M_2$ are infinite.
a) $7 \in M_1, 10 \in M_2$;
b) sets $M_1$ and $M_2$ are infinite.