Prime numbers are far more interesting than I ever realised. When I was first introduced to them I really didn't see what all the fuss was about; and its taken many years to gradually get a very rudimentary idea of some of their awesomeness...
So you probably know that a prime number is one which only has two factors: 1 and itself. In other words, it can't be broken down into smaller numbers. An example of a prime number is 23.
When you compare 23 to 24, a composite (non-prime) number, you get two totally different pictures. There are lots of ways of breaking 24 down into smaller pieces: 8x3, 6x4, 12x2 are some examples. If you break it down into the very smallest possible pieces (in fact, into prime pieces or factors) you end up with 2x2x2x3.
By contrast, 23 can't be broken down into smaller factors at all!
Of course 23 is quite a boring example. But when you find an extremely large prime number, like 243,112,609 − 1 (which has 12,978,189 digits) well then the fact that it has absolutely no factors other than 1 and itself is really quite impressive.
According to Charlie Epps, this has amazing applications in computer coding. According to Scarlett Thomas (in PopCo, which you HAVE to read immediately if you haven't already) it has exciting applications in codes as in secret codes.
From my extremely lay-person's point of view, it seems the key to this is that when you multiply two very large prime numbers together to make an even larger composite number, it is very difficult to work backwards and work out what those original two prime numbers were. Hence some serious encryption...
However, if like me you don't really care too much about the details of applications, it's still incredible to think of such an enormous number, and then to try get your mind around the fact that it is not made up of any smaller pieces.