The most famous problem involving prime numbers is Goldbach's Conjecture, which is actually several conjectures. In 1642, Christian Goldbach wrote a letter to my personal math man crush Leonhard Euler making several conjectures. The one that is now called Goldbach's Conjecture is stated as follows.
Every even number greater than 2 can be written as the sum of two primes.
Technically, this is the strong Goldbach's Conjecture. The weak conjecture is this.
Every odd number greater than 7 can be written as the sum of three odd primes.
Why the second one is weak and the first one strong is that if the second one is true, the first one might not be true, but the strong conjecture would prove the weak conjecture, since if any even number greater than 2 is the sum of two primes, add three to that sum and you get an odd number that is at least 7, and 7 only works as 2+2+3. Back in Goldbach's day, 1 was a prime but it isn't any more, so he could have written 1+1+5 or 1+3+3 as three primes adding up to 7.
If we find one even number that doesn't work, Goldbach's Conjecture is false. People have tried. Back in 1938, a guy named Pipping checked up through 100,000 by hand. Nowadays with computers, a program designed by T. Oliveira e Silva has checked up through 10^18, which is a million trillion. This method can never prove Goldbach's Conjecture to the satisfaction of mathematicians, but it could possibly disprove it.
Like a lot of number theory, there isn't a practical use if we find Goldbach's Conjecture to be true or false. Mathematicians are drawn to problems that can be easily stated, and while the fact that this one has been lying around unsolved for 267 years should be a red flag that it must be hard to solve, that doesn't stop people from trying. After all, Andrew Wiles' proof of Fermat's Last Theorem came over 350 years after Fermat wrote that he knew how to prove it but didn't write it down, which meant it was a conjecture and not a theorem for all that time.