Competition problem — Number theory

Number theory Difficulty 4.3 AIME Find the answer Nordic Mathematical Olympiad

Problem:
Find one solution in positive integers to the equation
$x^{2} - 2x - 2007y^{2} = 0$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
The equation can be written in the form
$x(x - 2) = 223 \cdot (3y)^{2}$
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 $223 \cdot (3y)^{2}$ for $y = 5$ . Thus, $(x, y) = (225, 5)$ is one solution.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.