BALKAN 2021 Problem 3

Let $a, b$ and $c$ be positive integers satisfying the equation

$$(a, b)+[a, b]=2021^{c} .$$

If $|a-b|$ is a prime number, prove that the number $(a+b)^{2}+4$ is composite. Here, $(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $[a, b]$ denotes the least common multiple of $a$ and $b$.