Prime Decomposition of 5th Fermat Number/Proof 2

Proof
Note the remarkable coincidence that $2^4 + 5^4 = 2^7 \cdot 5 + 1 = 641$.

First we eliminate $y$ from $x^4 + y^4 = x^7y + 1 = 0$:

Now we use the above result for $x = 2$ and $y = 4$ in modulo $641$:

Thus $2^{\left({2^5}\right)} + 1 = 6700417 \times 641$ and hence is not prime.