Mathematical Reflections 2015, Issue 3 - Problem J337

Problem:
Prove that for each integer $n \geq 0$, $16^n + 8^n + 4^{n+1} + 2^{n+1} + 4$ has two factors greater than $4^n$.

Proposed by Titu Andreescu, University of Texas at Dallas

Solution:
We have $$\begin{array}{lll} 16^n + 8^n + 4^{n+1} + 2^{n+1} + 4&=& (16^n+4^{n+1}+4)+2^n(4^n+2)\\&=&(4^n+2)^2+2^n(4^n+2)\\&=&(4^n+2)(4^n+2+2^n), \end{array}$$ and it's clear that each factor is greater than $4^n$.