1. Solve the equation in integers

$$
2025^{x}-100 x y+3-y^{2}=0
$$
Answer: $(0 ; 2),(0 ;-2)$.

## Solution.

Consider $x, y \in Z$.

$$
2025^{x}-100 x y+3=y^{2} .
$$

1) $x=0 \Rightarrow 2025^{0}-0+3=y^{2} \Rightarrow 4=y^{2} \Rightarrow y= \pm 2$.
2) $x \neq 0 \Rightarrow 2025^{x}-100 x y+3=y^{2} \Rightarrow$ no solutions, because the number on the left side ends with the digit 8, while the number on the right side is a perfect square.