<s>[INST] How to prove $2^{2^{6n+2}} + 21$ is a composite number? How to prove that $$ 2^{2^{6n+2}} + 21 $$ is a composite number for each $$n=1,2,3,\ldots$$
I'm stuck on a what seems like a simple problem but I don't even know where to start. 
 [/INST] $ 2^{2^{6n+2}} + 21=2^{4 (2^{6n} )}-2^4+2^4+21= $
$ =37+2^4\left (2^{4(2^{6n}-1 )}-1 \right )=37+2^4\left (2^{4(2^6-1 )(2^{6(n-1)}+ \dots +1 )}-1 \right )=$
$ =37+2^4\left (2^{36 \cdot 7 \cdot (2^{6(n-1)}+ \dots +1 )}-1 \right )=$
$ =37+2^4\left (2^{36}-1\right )\left (\dots   \right ) \equiv 0 \bmod 37$
</s>