Track / Stage 4 / 191 of 340 #451 of 1964

Problem 451

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

Determine all solutions $(x, y) \in \mathbf{N}^{2}$ of the equation:

$x(x+1)=4 y(y+1)$

Your answer

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

Official solution

$(x, y)=(0,0)$ is obviously a solution. Furthermore, the equation can be written as: $(2 y+1)^{2}=$ $4 y(y+1)+1=x^{2}+x+1$ , for $x>0, x^{2}+x+1$ , strictly between $x^{2}$ and $(x+1)^{2}$ , cannot be a perfect square.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.