[pilarphosol]

Prime numbers are interesting in this regard not for the notional reason (low Kolmogorov complexity) but because if their high “randomness.” They are not patterned so much as they are the leftovers excluded by patterns (notably, nontrivial multiplications) but, for this reason, of interest to number theorists simply because facile pattern “shouldn’t be” there. Which I guess gets to the heart of why diagonal arguments so easily torpedo discussions of “interestingness” or the lack thereof.

[yrdmb]

One of my favorite stories concerning primes is how fermat thought he discovered the formula for primes ( Fermat's primes ). He only checked his formula for the first few primes and assumed it worked for all primes. Euler checked for the next prime using fermat's formula and discovered the number to be composite and proved fermat's formula didn't work.

[pilarphosol]

It’s worse than that, even, because it’s believed there are no Fermat primes at all after 65,537. It’s one of the few cases of a conjecture being (with very high probability) 100 percent wrong.