Question 54
We say that a triple of integers $(x, y, z)$ is *nice* if $x$, $y$ and $z$ are positive integers, with $y \geq 2$, and $x^{2}-3 y^{2}=z^{2}-3$.
a) Find a nice triple $(x, y, z)$ with $x=5$ and one with $x=7$.
b) Show that for every odd $x \geq 5$ there exist at least two distinct nice triples $\left(x, y_{1}, z_{1}\right)$ and $\left(x, y_{2}, z_{2}\right)$.
c) Find some nice triple with $x$ even.
a) Find a nice triple $(x, y, z)$ with $x=5$ and one with $x=7$.
b) Show that for every odd $x \geq 5$ there exist at least two distinct nice triples $\left(x, y_{1}, z_{1}\right)$ and $\left(x, y_{2}, z_{2}\right)$.
c) Find some nice triple with $x$ even.