User:ArlenCuss/Primes


 * Suppose there are finite primes: $$\mathbb{P} = \{ p_1, p_2, p_3, ..., p_n \}$$
 * Let $$p = \prod_{i=1}^{n} p_i + 1$$
 * $$\therefore p > p_i \forall i \in [1, n]$$
 * $$\therefore p \notin \mathbb{P}$$
 * $$\therefore p$$ is not prime
 * $$\therefore \exists i \in [1, n] : p_i \mid p$$
 * But, $$p \equiv 1 \mod p_i \forall i \in [1, n]$$
 * $$\therefore$$ primes are infinite.