WEBVTT

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Hi, yesterday we saw something remarkable. Prime

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numbers never run out. No matter how far you

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go, there will always be another. So, a natural

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question follows. If primes go on forever, where

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are they? How often do they appear? At first,

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you might expect a pattern. After all, these

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numbers are not random. They are completely determined.

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A number is either prime or it isn't. There is

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no guesswork. And yet, when you start listing

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them, 2, 3, 5, 7, 11, 13, they do not feel like

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a pattern. Sometimes they're close together,

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11 and 13. Other times they spread out. There's

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a gap between 23 and 29. And as numbers get larger,

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those gaps tend to grow. Primes become more rare.

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So now, we're in a rather interesting place.

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We know they go on forever, but we do not see

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a simple rule telling us exactly where the next

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one will be. It's like looking at something that

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is perfectly ordered, but appears almost random.

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And that tension has fascinated mathematicians.

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for centuries. Over time, we've learned something

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that, at least to me, is surprising. Even though

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we cannot predict exactly where the next prime

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will be, we can begin to understand how often

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they appear. Not exactly, but approximately.

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As numbers grow larger, primes thin out in a

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very consistent way. There's a kind of hidden

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rhythm to them. Not a simple pattern you can

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see at a glance, but a deeper structure that

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emerges over time. So primes live in this strange

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space. Completely determined, yet difficult to

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predict. Structured, yet mysterious. And maybe

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that is part of their beauty. Not everything

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that is real is immediately obvious. Sometimes

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understanding does not come from finding a simple

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rule, but from learning how to see differently,

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from stepping back, from gaining altitude. So

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today, as you think about prime numbers, hold

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on to that tension. Something can be fully determined

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and yet still feel unpredictable. And tomorrow,

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we will go one step further. and look at just

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how much we still do not understand.