2 Took the +1 out of the exponent

Here's another one, maybe mostly of historical interest. Fermat once conjectured that all numbers of the form $$p=2^{(2^n)+1} p=2^{2^n}+1$$ are prime, which he had the means to verify up to $n=4$. It took more than 100 years until Euler showed that this fails at $n=5$. Today we still don't know if there are any other Fermat primes, so quite possibly Fermat's conjecture fails in the worst possible way.

Here's another one, maybe mostly of historical interest. Fermat once conjectured that all numbers of the form $$p=2^{(2^n)+1}$$ are prime, which he had the means to verify up to $n=4$. It took more than 100 years until Euler showed that this fails at $n=5$. Today we still don't know if there are any other Fermat primes, so quite possibly Fermat's conjecture fails in the worst possible way.