Page 101 - 'Blast_Into_Math

Blast into Math! Prime nummers: indestructimle muilding mlocks

Since Q is the product of primes which are all at least as big as 2, Q> 1. So, Q ∈ N , Q> 1

and Q< n +1. By the induction assumption, the theorem is true for Q . So, the finite sequence
{P 2 ,...,P k+m } is unique up to being re-arranged. This means that whenever Q is equal to the
product of all the elements of a finite sequence of primes, that finite sequence is just {P 2 ,...,P k+m }
written in some order. Since

n +1 = P 1 ∗ Q,

by the Long Division Theorem, Q is the unique integer such that its product with P 1 is equal to n +1.
Therefore, the finite sequence S consists of P 1 together with {P 2 ,...,P k+m } in some order. But, this
is the same as {P 1 ,...,P k+m }. So, any finite sequence of primes whose product is n +1 is unique

up to being re-arranged.

♥

To help understand the theorem, let’s do some examples. For any prime number p, what is its unique
prime factorization? What is the finite sequence of prime factors of p ? By definition of prime, the only
positive integers which divide p are 1 and p . Since 1 is not prime, the only prime divisor of p is p ,
so its unique finite sequence of prime factors is {p}.

Next, let’s think about composite numbers.

Exercise: What is the smallest composite natural number?

What is the prime factorization of the smallest composite natural number?

4= 2 ∗ 2.

So, the finite sequence of prime factors of 4 is the set {2, 2} which is the set containing the number 2
two times. What is the next composite number? What is its unique prime factorization?

6= 2 ∗ 3.

The prime factors of 6 are 2 and 3. So, the finite sequence of primes {2, 3} and the finite sequence of
primes {3, 2} both satisfy the theorem. But, these finite sequences are the same up to being re-arranged.

Exercise: Pick a composite number n and follow the script for the proof of the FTA. Audition the actors
and play out the script until you really understand the FTA. Don’t forget to applaud!

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