I'm having trouble with this proof: Prove that there are infinitely many prime numbers, using proof by contradiction. My teacher gave it to us the first day of class, but I'm a little lost: Assume there are finitely many prime number, P1, P2, P3,.....,Pn. Let q=(P1*P2*P3*.....*Pn)+1 Then, for all 1≤i≤n, Pi does not divide q, so therefore there must be at least Pn more prime, Pn+1. Here we reach a contradiction since we assumed there were only n primes. Therefore our assumption is in error and there are infinitely many primes. # I don't understand the portion in red. Can anyone please explain it in simple terms? Thanks.