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 $x^4 + y^4 = x^7 y + 1 = 0$:

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

Thus $2^{\paren {2^5} } + 1 = 6 \, 700 \, 417 \times 641$ and hence is not prime.