Question 8
Let $p \neq q$ be coprime positive integers. Determine all infinite sequences $a_1, a_2, \ldots$ of positive integers such that the following conditions hold for all $n \geq 1$:
$$ \begin{aligned}\max(a_{n},a_{n+1},\cdots,a_{n+p})&-\min(a_{n},a_{n+1},\cdots,a_{n+p})=p\qquad \text{and}\\\max(a_{n},a_{n+1},\cdots,a_{n+q})&-\min(a_{n},a_{n+1},\cdots,a_{n+q})=q.\end{aligned} $$
$$ \begin{aligned}\max(a_{n},a_{n+1},\cdots,a_{n+p})&-\min(a_{n},a_{n+1},\cdots,a_{n+p})=p\qquad \text{and}\\\max(a_{n},a_{n+1},\cdots,a_{n+q})&-\min(a_{n},a_{n+1},\cdots,a_{n+q})=q.\end{aligned} $$