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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.