2021 IMO Shortlist N1

Determine all integers $n \geq 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and $$\frac{a b+3 b+8}{a^{2}+b+3}=n$$

确定存在一对正整数 $(a, b)$ 的所有整数 $n \geq 1$，使得素数的立方不能整除 $a^{2}+b+3$ 和 $$\frac{a b+3 b+8}{a^{2}+b+3}=n$$