Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\]with $|a - b|$ and $(a+b)^2 + 4$ are both prime number
Source: Indonesian TST for IMO 2023 stage 1: Test 3 - Number Theory
Tags: number theory
Find all triplets natural numbers $(a, b, c)$ satisfied \[GCD(a, b) + LCM(a,b) = 2021^c\]with $|a - b|$ and $(a+b)^2 + 4$ are both prime number