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The Fundamental theorem of Arithmetic
A natural number p > 1 is called prime if its only divisors are 1 and p.
With one exception, that of the number 2, all prime numbers are odd. The
distribution and the properties of prime numbers can be thought of as the
quintessential problem of mathematics. If a natural number is not prime
and it is greater than 1, then it will be called composite. In this note we
will study some basic properties of prime numbers.
Two integer numbers a and b are called coprime or relatively prime if
gcd(a, b) = 1
In this case, there exist x, y ∈ Z such that ax + by = 1.
Conversely if for two integer numbers a and b, one can find x, y ∈ Z such
that ax + by = 1, then a and b are coprime. Indeed if d is their greatest
common divisor, then d has to also divide ax + by, i.e. it has to divide 1,
i.e. it has to be equal to1.
We now proceed with a very useful theorem:
Theorem: Suppose that n ∈ N and a, b ∈ Z with gcd(n, a) = 1. If n|ab,
then n|b.
Proof: Since n|ab, there exists k ∈ Z such that:
ab = nk
On the other hand, since gcd(n, a) = 1, there exist x, y ∈ Z such that:
nx + ay = 1
If we multiply both sides by b, we get:
nbx + aby = b
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which may be rewritten as:
nbx + nky = b
or
n(bx + ky) = b
This implies that n|b, as desired.
Note that a prime number p either divides a given integer n, or it is coprime
to it.
In the sequel we will need a very important property of the natural numbers,
the so called well ordering principle:
Well ordering principle: Every nonempty subset of the natural numbers
has a least element.
Let’s see a simple application of the above principle:
Theorem: If a prime number divides a product of prime numbers, then it
has to be equal to one of them.
Proof: To prove this, we are going to argue by contradiction. We suppose
that there are products of prime numbers divisible by a prime numbers that
do not coincide with any of the prime factors. If this is true, then there
has to be such a product that has the least amount of prime factors. More
precisely, there are prime numbers q1 , . . . , qk and a prime number p, so that:
p|q1 · · · qk
and p does not equal to any of the qi ’s. More over we suppose that k is the
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least natural number for which something like this is possible. Since p 6= qk
and both of them are prime numbers, we have that
gcd(p, qk ) = 1
By the theorem that we proved above, p has to divide q1 · · · qk−1 and it is
not equal to any of the qi ’s. But this contradicts the fact, that k was the
least number for which something like that was possible - indeed we just
found that it is possible for k − 1. This establishes the desired contradiction
and completes our proof.
We will also show that any natural number greater than 1 is either prime
or it is a product of prime numbers. We will again use the well orderin
principle and argue by contradiction. Indeed, if our claim is not correct and
there are natural numbers that are greater than 1 and are neither prime
nor products of prime numbers, then there has to exist a smallest natural
number with this property. So let n be the smallest number (greater than
1) which is neither nor a product of prime numbers. Since it is not a prime,
it has to be composite. This means that there are natural numbers k and l
such that:
n = kl
with k > 1 and l > 1. But k and l have to be smaller than n and therefore
each one is either a prime number or a product of prime numbers. However
this implies that their product n will also be a product of prime numbers
which is a contradiction to the assumptions on n. This contradiction implies
that every natural number greater than 1 is either a prime or a product of
prime numbers.
We will finish this note, with a result that combines everything that we have
seen so far:
Fundamental Theorem of Arithmetic:Let n be a natural number greater
than 1. Then there exist unique prime numbers:
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p1 < . . . < ps
(it is possible that s = 1) and unique natural numbers k1 , . . . , ks so that:
n = pk11 · · · pks s
The essence of the above statement is that every natural number greater
than one can be factorized in a unique way as a product of prime numbers.
This unique factorization will be referred to as the prime factorization of
the given natural number.
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