Library / /540 of 1394

, 2020

Number theory Difficulty 5.2 AIME, harder Prove it United States

Problem:

Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3x^{2} + 10xy + 3y^{2}$ is the power of some prime.

Solution

Solution:

We can factor as $(3x + y)(x + 3y)$ . If $x \geq y$ , we need $\frac{3x + y}{x + 3y} \in \{1, 2\}$ to be an integer. So we get the case where $x = y$ , in which we need both to be a power of $2$ , or the case $x = 5y$ , in which case we need $y$ to be a power of $2$ . This gives us $11 + 9 + 9 = 29$ solutions, where we account for $y = 5x$ as well.

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