BdMO Online Forum

Consider two cases:
1. $n$ even. Sophie is inviting you!
2. $n$ odd. My old good old friend FLT.

for $n=2k$, $n^4+4^n$ is divisible by $2$
for $n=2k+1$

\[\begin{align*}
n^4+4^n &=(2k+1)^4+4^{2k+1} \\
&=(2k+1)^4+4\cdot4^{2k} \\
&=(2k+1)^4+4\cdot2^{4k} \\
&=(2k+1)^4+4\cdot(2^k)^4
\end{align*}\]

sophie germain