2, 3, 5, 7, 11, 13, 17, 19...
Where Prime Numbers mysteries are solved.
There are many many different types of prime numbers. Wikipedia has a long list of them here. Here are some of the more common types:
Mersenne primes are primes of the form Mp = 2p - 1. Mersenne primes have been around for a long time... they were even considered by Euclid in Ancient Greece. One fun fact about them, a Mersenne number may only result in a Mersenne prime when you use a prime number as an exponent! If you use a composite number as the exponent, it's been proven the resulting number will not be prime. So far we've found 47 Mersenne primes, but it's been conjectured there are an infinite number of them! The largest one found to date is equal to 243,112,609 - 1. It has 12,978,189 digits. Now that's a large number!
Twin primes exist whenever two primes have a difference of 2. So if p is prime, and p ± 2 is prime, that constitutes a prime pair. A few examples of prime pairs are 3 and 5, 5 and 7, and 11 and 13. The largest known twin primes are 3756801695685 x 2666669 ± 1. These numbers have 200,700 digits! The twin prime conjecture states that there are infinitely many primes p such that p + 2 is also prime. However, this is only a conjecture, it's not known whether or not it's true! Maybe that's one conjecture you'll someday prove or disprove!
Primorial primes are a little harder to explain than the other types above. It makes sense that a good way to find prime numbers would be to multiply prime numbers together, and then add or subtract one. Why? Because you know the resulting number won't be divisible by any of the primes you just multiplied together! For example, let's take the 3rd primorial numbers, which are made up by multplying the first 3 prime numbers together and adding or subtracting 1. These numbers are 2 x 3 x 5 ± 1, which equals 29 and 31. Both of these are prime! The largest known primorial prime today is the 843,301ist primorial number (subtracting 1), which has 365,851 digits!