Fundamental Theorem of Arithmetic:
Every integer greater than one is either prime or is a product of primes.
Proof.
For each positive interger n > 1, let S(n) be the statement “n is a prime or is a product of
primes”.
1. S(2) is true since 2 is a prime number.
2. Let k be any integer, k ≥ 2. We will show that if S(2), S(3), … , S(k) are true,
then S(k+1) is true. If k+1 is a prime number, then S(k+1) is obviously true.
If k+1 is not a prime number then k+1 = a b where a and b are integers,
2 ≤ a ≤ k, 2 ≤ b ≤ k. Now if S(2), S(3), … , S(k) are true, then S(a) and S(b)
are true. So, both a and b are either prime or the product of prime numbers.
Hence, k+1 = a b can be written as a product of primes.
By the principle of strong induction, S(n) is true for all n ≥ 2.