Name: Squaring Primes - Numberphile
Uploaded: 2020-03-27T18:21:54.000Z
Duration: 13 min 48 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Future simple

today, we're gonna get the fact that all prime numbers when you square them are one more than a multiple of 24 which I love him.

When I first told you about this, you're you're beside yourself because people think the prime numbers haven't got a pattern.

Like a lot of emails from people saying, Oh, I found a pattern.

The prime numbers on their load, the patterns in the prime numbers on dhe, including this one which will test So, Brady, what prime number should we do with the black Sharpie?

Let's do 17 7 Okay, 17 squared equals about.

OK, so it's gonna be one of the small No, I can do this cause Square's gonna be 100 and 70 plus seven times that, which is going to be 70 plus 49.

Okay, so just off the top of my head, uh, that was that.

Okay, I think it's roughly, uh, that's not right.

And so I'm saying that's one more than a multiple off 24 and so we can split that apart because we've got 240 hidden in there, plus 49 left over.

And that's one more than 48 which is a multiple of 24.

And so this whole thing here is gonna be at 12 times 24 plus one.

So I argue, any prime number you give me If you square it, you'll get some multiple of 24 plus one on the end.

But no, we do 17 1st It doesn't work if you go back up and do two or three, so two or three don't work.

And I mean, I argue that they're not real prime numbers.

I called in the sub primes, so I would like to ignore those for the purposes of this.

This always works, but we should prove it right.

First of all, I was amazed that I was like, what?

There must be a reason why there's this pattern.

There are loads of patterns in the prime numbers.

So there's a nice one involving multiples of six.

And because 24 is a multiple of sex was like, You know what?

So I'm gonna very quickly list out a number.

We got the subprimes hanging out down here and then we got the real times.

11 13 17 19 on someone and you get right Where the multiples off.

Look that there's a prime on either side of it, right?

So those two either side of Mobile six There's the next multiple of six.

So the primes are always above and below all the multiples of six, except doesn't always work.

So if I kept going the number Liar stopped at 19.

There is our next multiple of six, and only on one side do we get a prime.

25 is not up until now, they have all been, and the reason this one isn't is five finally caught up with us, so five is prime, but then every fifth number isn't tends.

The moral of story is not that there's something magical about the primes, and they happen to always be above and below every multiple.

It's just that's the only place they can be.

So here's a multiple of six, and here's the next one.

They can't be the even points in between, because we know primes can't be even.

It knocks out these two, and it can't be the number in between because a multiple of three number between multiple of sixes, multiple three.

So it can't be on any of these, which is why it has to be above or below a multiple of six.

It's just a fancy way of saying primes don't have two or three as a factor.

And then, to their constrained to those and so people are happy.

If you say all primes are odd ignoring this moment, right?

And that's just because this is nobody's saying primes haven't got two is a factor.

If you say all primes are one Maur or less, the mobile of six was like, Wow, but always saying is they haven't got three is a factor.

And then when we say all prime squared give you one more than 24 it's a variation on this.

And so the way actually, the way I first worked it out when I came across, it feels like I'm going to prove this delivery.

Here's what I did I said any prime number other than two or three is either gonna be some most pool of six plus one or some multiple of six minus one.

So every K we can put it on these two categories.

And so my thought was, I just square these and sure, they're both one more, one less than a multiple off 24.

But then they didn't work out really, really complicated.

So I had another cheating moment when I realized this kay here is either gonna be odd or even so that Kay is either gonna equal that's used.

She was m this time to em, or it's going to equal this is, or two em plus once either even or odd.

And so I can split each of these into their two options.

So this one is either gonna be If I put to m in there is gonna be 12 m plus one or if I put in to M plus one, it's gonna be 12 m plus seven and then down here that's either going to be put in to end.

That's gonna be 12 m minus one, for it's gonna be 12 m plus 61 5 Okay, so now I know every single prime number falls into one of these four categories.

And then I went through and I took each of these and, uh, squared them to see what happens.

I've just taken all the options and put them into four categories, and now I'm going to check each of the categories separately.

Plus two times that time this one's 144 em.

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Squaring Primes - Numberphile

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