(related to Theorem: Infinite Set of Prime Numbers)

- Assume, there are only finitely many prime numbers \(p_1, p_2,\ldots,p_r\).
- Calculate the number $n:=p_1\cdot p_2\cdot \cdots \cdot p_r+1.$
- By assumption \(n\) is composite.
- However, note that $n$ cannot be divisible by any of the primes \(p_1, p_2,\ldots,p_r\), otherwise it would follow from the divisibility law no. 6 that also 1 is divisible by this prime number, which is impossible.
- Therefore, there has to exist another prime dividing \(n\), in contradiction to our assumption.∎