What are some interesting facts about prime numbers?

Prime numbers are truly fascinating entities in the world of mathematics. Can you tell me, are there any particularly intriguing facts about them that stand out to you? For instance, did you know that there are an infinite number of prime numbers, yet we still haven't found a definitive formula to predict the next one? Or how about the fact that the largest known prime number is constantly being surpassed by even larger ones, thanks to advancements in computing power? I'm eager to learn more about these enigmatic numbers and their remarkable properties.