$F_{5}=2^{2^{5}}+1 $ is not prime

Euler made this awesome computation :

We have $641 = 1 + 5 \times 2^7$. Therefore $5 \times 2^7 \equiv -1 \; [641]$. By squaring twice this congruence, we get : $5^4 \times 2^{28} \equiv 1 \; [641]$. However we also have $5^4 + 2^4= 641$. Therefore, $ - 2^4 \times 2^{28} \equiv 1 \; [641]$, which yields $ 1 + 2^{32} \equiv 0 \; [641]$. And since $32 = 2^5$, you are done showing that $F_5$ is not prime. Long live Euler's computations...