The mathematical pattern seemed so elegant. 21+1=3, 22+1=5, 24+1=17... all prime numbers. Even 28+1=257 and 216+1=65537 are prime. So naturally, one might conclude that all numbers of the form 22n+1 are prime.
Except they're not. This is the Fermat prime conjecture trap. Fermat numbers F5 and beyond are actually composite. F5 = 232+1 = 4,294,967,297 = 641 × 6,700,417.
Mathematics: where induction from a few examples will make you look like that third panel. Number theory doesn't care about your feelings or your pattern-seeking brain.