If There Exist Only Two Primes, Then…

There must exist at least two prime numbers. If 2 is the first prime number, then 3 must also be a prime because no two consecutive integers have the same prime factorization. If 2 and 3 were the only prime numbers then the rest of the integers would be either the power of 2 or 3.  The set of positive integers would appear like

S = {2, 3, 2*2, 3*3, 2*2*2, 3*3*3,….}.

The mapping f: S –> N would imply

2 = 2, 3 =3, 2*2 = 4, 3*3 = 5, 2*2*2 = 6, 3*3*3 = 7, etc.