If you want a less complex unsolved problem involving primes, here are a couple of good ones:
The Goldbach Conjecture: Every even number greater than 2 can be expressed as the sum of two prime numbers. For instance, 4 = 2+2, and 110 = 107 + 3 (many numbers can be broken down multiple ways). It’s been tested up to absurdly high numbers, and it’s always worked out, but nobody’s ever been able to come up with a general proof, and many folks suspect that a general proof is impossible.
The Twin Prime Conjecture: A pair of twin primes is just two primes that are separated by 2, such as 5 and 7, or 107 and 109. The Twin Prime Conjecture quite simply says that there are an infinite number of such pairs. Again, it’s widely suspected to be true, but there’s even less progress on proving it than there is for the Goldbach conjecture.