
07.1. Find one solution in positive integers to the equation

$$
x^{2}-2 x-2007 y^{2}=0
$$


Solution. The equation can be written in the form

$$
x(x 2)=223 \cdot(3 y)^{2}
$$

Here the prime number 223 must divide $x$ or $x 2$. In fact, for $x=225$ we get $x(x 2)=$ $15^{2} \cdot 223$, which is equivalent to $223 \cdot(3 y)^{2}$ for $y=5$. Thus, $(x, y)=(225,5)$ is one solution.
