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Blast into Math! Prime nummers: indestructimle muilding mlocks

These sequences are unique up to re-arrangement. This means that if x is equal to the product of a
different finite sequence of primes, then that sequence is just the same list {p 1 ,p 2 ,...,p k } written in
a different order. Similarly, if y is equal to the product of a different finite sequence of primes, then that
sequence is just the same list {q 1 ,p 2 ,...,p m } written in a different order. So, we know that

n +1 = xy = p 1 ∗ p 2 ∗ ... ∗ p k ∗ q 1 ∗ q 2 ∗ ... ∗ q m .

We have found a finite sequence of prime numbers,

k+m
{P j } j=1 , P j = p j for1 ≤ j ≤ k, P k+i = q i for1 ≤ i ≤ m.

This means that the first k elements in the list are {p 1 ,...,p k } , and the next m elements in the list
are {q 1 ,...,q m } . To complete the proof of the theorem, we must show that this finite sequence is
unique up to being re-arranged. Since P 1 |(n +1), by the CCP, if we write n +1 as the product of a

finite sequence of primes, P 1 must be contained in that finite sequence. So, any finite sequence of primes
whose product is n +1 contains P 1 . If S is a finite sequence of primes whose product is n +1, then
we know that P 1 is an element in S . Since P 1 is prime, P 1 ≥ 2, and

k+m

Q = P j <P 1 ∗ Q = n +1.
j=2

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