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Imagine we are living in prehistoric times.

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Now, consider the following:

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How did we keep track of time without a clock?

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All clocks are based on some repetitive pattern

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which divides the flow of time into equal segments.

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To find these repetitive patterns,

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we look towards the heavens.

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The sun rising and falling each day

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is the most obvious [pattern].

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However, to keep track of longer periods of time,

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we looked for longer cycles.

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For this, we looked to the moon,

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which seemed to gradually grow

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and shrink over many days.

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When we count the number of days

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between full moons,

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we arrive at the number 29.

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This is the origin of a month.

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

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we run into a problem: it is impossible.

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The only way to divide 29 into equal pieces

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

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29 is a 'prime number.'

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Think of it as unbreakable.

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If a number can be broken down into

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equal pieces greater than one,

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we call it a 'composite number.'

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Now if we are curious, we may wonder,

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"How many prime numbers are there?

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– and how big do they get?"

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

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We list the primes on the left

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and the composites on the right.

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At first, they seem to dance back and forth.

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There is no obvious pattern here.

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So let's use a modern technique

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to see the big picture.

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The trick is to use a "Ulam spiral."

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First, we list all possible numbers in order

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in a growing spiral.

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Then, we color all the prime numbers blue.

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Finally, we zoom out to see millions of numbers.

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This is the pattern of primes

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which goes on and on, forever.

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Incredibly, the entire structure of this pattern

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is still unsolved today.

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We are onto something.

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So, let's fast forward to

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around 300 BC, in ancient Greece.

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A philosopher known as Euclid of Alexandria

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understood that all numbers

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could be split into these two distinct categories.

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He began by realizing that any number

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can be divided down – over and over –

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until you reach a group of smallest equal numbers.

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And by definition, these smallest numbers

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are always prime numbers.

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So, he knew that all numbers are

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somehow built out of smaller primes.

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To be clear, imagine the universe of all numbers –

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and ignore the primes.

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Now, pick any composite number,

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and break it down,

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and you are always left with prime numbers.

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So, Euclid knew that every number

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could be expressed using a group of smaller primes.

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Think of these as building blocks.

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No matter what number you choose,

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

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This is the root of his discovery,

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known as the 'Fundamental Theorem of Arithmetic' –

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as follows:

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Take any number – say 30 –

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and find all the prime numbers

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it [can be divided into] equally.

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This we know as 'factorization.'

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This will give us the prime factors.

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

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Euclid realized that you could then multiply

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these prime factors a specific number of times

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to build the original number.

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In this case, you simply

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multiply each factor once to build 30.

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

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Think of it as a special key or combination.

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There is no other way to build 30,

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using some other groups of prime numbers

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multiplied together.

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So every possible number has one –

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and only one – prime factorization.

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A good analogy is to imagine each number

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as a different lock.

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The unique key for each lock

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would be its prime factorization.

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No two locks share a key.

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No two numbers share a prime factorization.