Primes look like they follow no pattern, but surprising patterns hide everywhere within them. Some of these patterns "seem clearly true," yet mathematicians still haven't managed to prove them. Try plugging in numbers yourself and meet these famous unsolved problems.
Twin primes are pairs of primes exactly 2 apart, like (11,13). Goldbach's Conjecture claims that every even number greater than 4 can be written as the sum of two primes. Fermat primes are the prime numbers among 2 raised to a power of 2, plus 1. All three always seem to hold whenever you check them, yet no one has ever proven "it's always true" (and for Fermat primes, the opposite is also unproven — only 5 have ever been found, but no one has proven there isn't a 6th).
Prime pairs with a difference of exactly 2, like (3,5) or (11,13), are called twin primes. Primes get rarer and rarer as numbers grow larger, yet twin primes just keep turning up.
This conjecture claims that every even number greater than 4 can be made by adding two primes. Since it was first proposed in 1742, computers have checked every even number up to 4×10¹⁸, but there's still no proof for "every even number."
A number of the form 2^(2ⁿ)+1 is called a Fermat number. Fermat thought these numbers would always be prime, but starting at n=5, they turned out not to be. Only 5 have ever been confirmed prime so far.