Remember those fancy prime numbers learnt in school? They're weird. First the set of numbers begins with $2$. Next it's $3$, $5$, $7$, $11$, $13$, $17$.... and there is simply no just pattern! Not only is there no pattern - but there are an infinite number of them - and this is what is post will prove. Why am I doing this? Why not! A small view into the infinite scale on a finite definition of numbers. We begin with the claim that there in fact exists a finite set of prime numbers. Call them $$p_{1}, p_{2}, \cdots, p_{n}.$$ With such a finite set - let's now call $$N = p_{1}\times p_{2}\times\cdots \times p_{n}+1.$$ Now $N$ is not prime since it's not in the finite set defined above, and $N$ is greater than 1. Furthermore, see that $$N-1 = p_{1}\times p_{2}\times\cdots \times p_{n},$$ is also not prime as it is divisble by every $p_{i}$ defined. However, even though we've made this claim - there can't exist a prime number that divides $N-1$ as $$\frac{N-1}{p_{i}}=\frac{N}{p_{i}}-\frac{1}{p_{i}},$$ and $p_{i}$ does not divide $1$. Therefore $N-1$ is in fact prime - and we have a contradiction in our original claim. Therefore - our original claim is false and there cannot exist a finite set of prime numbers. What's interesting here is that we can go through the prime numbers one by one, and of course, it will get trickier to find the next. However, from this simple proof we can be sure that there is an infinite number of them. Enough math for the night.