Was Fermat Wrong?

The question is bit $\color{#3D99F6}{\textbf{informative}}$ , we know that

$2^{2^n}+ 1$ is prime for $n\in$ {0,1,2,3,4} and for $n \in [5,32]$ , it has been proven that $2^{2^n}$ is NOT prime.

After a lot of research, finally i realized that it is $5$ . This proof was given by Euler.He stated that it if $5$ is put in place of n, then the resultant value, i.e., $4294967297$ = $641$ X $6700417$ , and that beats Fermat!

Note that $641 = 5^4 + 2^4 = 5\times2^7 + 1$ . Then $F_5 \; = \; 2^{32} + 1 \; \equiv \; 1 - 2^{28}\times5^4 \; \equiv \; 1 - (5\times2^7)^4 \; \equiv \; 1 - (-1)^4 \; \equiv \; 0 \pmod{641}$ and hence $F_5$ is not prime.

Is there any other way to solve this question irrespective of putting values one by one..??