What is the famous prime number problem?

Could you elaborate on the famous prime number problem? It seems to be a mathematical enigma that has intrigued scholars for centuries. Is it related to the infinite sequence of prime numbers and the search for a pattern or a formula to predict them? Or is it a question about the distribution of prime numbers within the natural numbers, and whether there's a way to determine if a large number is prime without performing exhaustive divisibility tests? Perhaps you could explain the significance of this problem in the broader context of number theory and cryptography, and discuss some of the attempts that have been made to solve it over the years.