1. Determine the integer numbers $x$ and $y$ for which $x^{2}-5^{y}=8$

Ovidiu Bădescu, RMCS Nr.40/2012
1. The equation is equivalent to $\quad x^{2}-8=5^{y}$
$x \in Z \Rightarrow x-8 \in Z$ and thus $\left.\quad 5^{y} \in Z \quad\right\} \Rightarrow y \in N$ ..... $1 p$
$\operatorname{but} y \in Z$
$U_{c}\left(8+5^{y}\right)=3$ for any $y \in N^{*}$ ..... $2 \mathrm{p}$
The given equation is impossible for any $y \in N^{*}$ ..... $.1 \mathrm{p}$
If $y=0$ then $x^{2}=9$
$x \in Z \quad\} \Rightarrow x \in\{-3,3\}$ ..... $.2 \mathrm{p}$
Finalization, $(x, y) \in\{(-3,0) ;(3,0)\}$ ..... $1 \mathrm{p}$