The Proof:

Suppose that there are only a finite number of prime numbers, say n of them. Let us label them p₁= 2, p₂ = 3, p₃ = 5, and so on, until p_n. Then, consider the following number

P = p₁p₂p₃... p_n + 1.

What are the prime factors of this new number? None of the n prime numbers we have is a factor! Because if we take any of the known prime, say p_i, and divide it into P, the result will be a quotient of p₁p₂...p_i-1p_i+1...p_n, and a remainder of 1. That is, none of the choices pi will exactly divide P. Therefore, either that the new number P is yet another prime number, or it is a composite number which has a whole new set of prime factors not from our original list of n prime numbers. In any event, at least one more prime number exists, which contradicts the assumption that there are only n (which can be an arbitrarily large finite number) prime numbers. Hence the number of prime numbers is infinite.

Next time: How are the prime numbers distributed among the integers - the Prime Number Theorem