Prove that there exists an infinitely large number of primes.
If possible let the total number of primes be finite and P be the largest prime.
Let F(P) = 2*3*.........*P, so that F(P) corresponds to the product of all possible primes< = P
Now, (F(P), F(P) +1) = 1, so that each of the prime factors of F(P) + 1 must be relatively prime to each of the prime factors of F(P).
Hence, it follows that each of the prime factors of F(P) + 1 must be different from all the known primes upto P.
This is a contradiction.
Consequently, it follows that there exists an infinity of prime numbers.