TITLE: Prove the sequence has infinitely many integers
QUESTION [5 upvotes]: Prove the sequence $a_n=\frac{p^n}{qn+1},(p,q)=1,p\ge 2$ has infinitely many integers 
Could someone explain the intuitive approach for this problem?

REPLY [3 votes]: Let $a$ the order of $p$ modulo $q$, hence $p^a$ is congruent to $1$ modulo $q$. Hence for $k\in \mathbb{N}$ we get $p^{ak}=1+m_k q$. Now $p^x>1+xq$ for $x$ large. Hence for large $k$, we have $m_k>ak$. Now put $n=m_k$, we get $\displaystyle \frac{p^n}{1+nq}=p^{m_k-ak}\in \mathbb{N}$.