Mathematical Reflections 2013, Issue 5 - Problem J277

Problem:
Is there an integer $n$ such that $4^{5^n}+ 5^{4^n}$ is a prime?

Proposed by Titu Andreescu.

Solution:
The answer is no. Clearly, for $n<0$ the given expression is not an integer. If $n=0$ we get $4+5=9$, which is not a prime. If $n>0$, set $x=5^{4^{n-1}}$ and $y=4^{\frac{5^n-1}{4}}$. It is easy to see that $x$ and $y$ are both integers. Hence $$4^{5^n}+5^{4^n}=x^4+4y^4=(x^2+2y^2+2xy)(x^2+2y^2-2xy),$$ which is the product of two positive integers greater than $1$.