The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic

Imagine we are living in prehistoric times.

Now, consider the following:

How did we keep track of time without a clock?

All clocks are based on some repetitive pattern

which divides the flow of time into equal segments.

To find these repetitive patterns,

we look towards the heavens.

The sun rising and falling each day

is the most obvious [pattern].

However, to keep track of longer periods of time,

we looked for longer cycles.

For this, we looked to the moon,

which seemed to gradually grow

and shrink over many days.

When we count the number of days

between full moons,

we arrive at the number 29.

This is the origin of a month.

However, if we try to divide 29 into equal pieces,

we run into a problem: it is impossible.

The only way to divide 29 into equal pieces

is to break it back down into [29] single units.

29 is a 'prime number.'

Think of it as unbreakable.

If a number can be broken down into

equal pieces greater than one,

we call it a 'composite number.'

Now if we are curious, we may wonder,

"How many prime numbers are there?

– and how big do they get?"

Let's start by dividing all numbers into two categories.

We list the primes on the left

and the composites on the right.

At first, they seem to dance back and forth.

There is no obvious pattern here.

So let's use a modern technique

to see the big picture.

The trick is to use a "Ulam spiral."

First, we list all possible numbers in order

in a growing spiral.

Then, we color all the prime numbers blue.

Finally, we zoom out to see millions of numbers.

This is the pattern of primes

which goes on and on, forever.

Incredibly, the entire structure of this pattern

is still unsolved today.

We are onto something.

So, let's fast forward to

around 300 BC, in ancient Greece.

A philosopher known as Euclid of Alexandria

understood that all numbers

could be split into these two distinct categories.

He began by realizing that any number

can be divided down – over and over –

until you reach a group of smallest equal numbers.

And by definition, these smallest numbers

are always prime numbers.

So, he knew that all numbers are

somehow built out of smaller primes.

To be clear, imagine the universe of all numbers –

and ignore the primes.

Now, pick any composite number,

and break it down,

and you are always left with prime numbers.

So, Euclid knew that every number

could be expressed using a group of smaller primes.

Think of these as building blocks.

No matter what number you choose,

it can always be built with an addition of smaller primes.

This is the root of his discovery,

known as the 'Fundamental Theorem of Arithmetic' –

as follows:

Take any number – say 30 –

and find all the prime numbers

it [can be divided into] equally.

This we know as 'factorization.'

This will give us the prime factors.

In this case 2, 3, and 5 are the prime factors of 30.

Euclid realized that you could then multiply

these prime factors a specific number of times

to build the original number.

In this case, you simply

multiply each factor once to build 30.

2 × 3 × 5 is the prime factorization of 30.

Think of it as a special key or combination.

There is no other way to build 30,

using some other groups of prime numbers

multiplied together.

So every possible number has one –

and only one – prime factorization.

A good analogy is to imagine each number

as a different lock.

The unique key for each lock

would be its prime factorization.

No two locks share a key.

No two numbers share a prime factorization.