Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$ Proposed by Shayan Talaei