Fermat Prime Conjecture

Conjecture
All numbers of the form $2^{\left({2^n}\right)} + 1$, where $n = 0, 1, 2, \ldots$ are prime.

This was postulated by Fermat.

Refutation
This was proved wrong by Euler.

Although true for $n = 0, 1, 2, 3, 4$, the conjecture fails for $n=5$.

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.

Also see

 * Fermat Prime