**Theorem**: There are infinitely many primes **Proof**: BWOC Suppose there are finitely many primes $p_{1},p_{2},p_{3}\dots,p_{n} \; \text{ for some } n\in \mathbb{N}.$Consider $p_{1}p_{2}\dots p_{n}+1= N$None of the $p_{i}$ divide $N$ since they leave a remainder $1$. So none of the $p_{i}$ are in the PPF of $N$, By by [[Fundamental theorem of arithmetic|FTA]], $\exists \text{primes not in } p_{1},\dots p_{n}$ which leads to a contradiction.