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