I don't believe you are shocked.

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.

Sorry.

78 12 13.

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

17 times 17 equals to eight.

Now, I was right.

I doubted myself.

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.

+55 25.

That's one more than 24.

You could have opened with that, Brady.

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.

They don't work everything.

Five onwards.

This always works, but we should prove it right.

You don't take my word for these things.

It comes down to when I first all this.

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

Hang on.

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?

That might be something to do with it.

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

Life.

11 12 13 18 19.

Blood over.

Okay, so we got those.

And then let's find our favorites.

The primes.

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

There's five there.

Seven there.

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

Six.

There's multiple of six there at six.

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.

Look at that.

There's a prime on both sides of it.

Nothing else in between.

Next Mobile six over there.

Look at that.

There's a primary excited it.

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.

So 23 is a prime.

25 is not a prime.

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.

Not fifteens, not twenties, not boom.

So it's not this one out.

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.

Six.

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.

So immediately.

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.

It just looks more impressive.

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.

We knew Square.

Now, this is not exhaustive.

Like exhaustive.

Prove her.

I've checked every single option.

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.

So let's do them quickly and squared.

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

Someone will correct me if I'm wrong.

A minus tunes at times plus 25.

Okay, so we have to do now is show that every single one of these is a multiple 24 plus one, and the first one's reason's straightforward cause.

That's a multiple off Ah, 24 because that's 12 squares that six times 24.

So we actually put 24.

Outside of that's gonna be sixth M squared, plus him plus one, said Local 24 plus one.

This one is 24 outside of again.

Six AM Squid plus seven mm plus two plus one.

So I've done there is at 48 5 49 I've realized, is 48 plus one.

And so the 2 24 to 48 plus the one on the outside and same deal again.

24 outside, 6 a.m. Squared minus one plus one.

Must them and finally 24 outside of six m squared plus five M plus one plus one.

So there I've taken every single prime showing that must be in one of two categories above or below multiple.

Six Each of those has to be one of two categories.

If that multiple sixes, even a rod.

Those full categories cover every single prime, then expanded them out and show that if you square them, you get some number times 24 Mobile 24 plus one, which proves it right.

It's a slightly clever, exhaustive proof where I put it in categories, and I've dealt with the category one of the time.

I was so pleased when I got to the end, it was like, Yep, I knew it had something to do with being one or more one less than a multiple of six.

I did some algebra.

I worked it out.

I showed some friends of mine, and one of them said, Why didn't you do it the easy way?

I was like, Ah, you know me.

I like to do some algae.

What easy way.

So it turns out there's an easier way to do this.

So I did this way.

This is mine.

I love it.

My friend Paul said, Look, all you're doing is you're looking at P squared, minus one and asking, Is that a multiple?

Off 24?

Every prime squared, subtract one is a Mobile 24.

Okay, P squared minus one.

You may remember this from school, or you could still be in school.

And you see that in a minute?

You think?

Well, that's a difference of two squares.

That's P minus one plus one.

It's been a while since you did this at school.

You could just double check that.

If you multiply these out, you'll get back to P squared minus one.

So what we really want to know is, is this a multiple of 24?

What can we say about this?

Well, on the number line that's going to go over here, we have the number one less than P.

P.

Myers one here.

We're gonna have the prime, he and then above it.

We're gonna the prime p.

Plus, Once we actually got three consecutive numbers here and the prime in the middle hasn't got any factors.

So we know every second number is a multiple of two.

So because these were in a row, we know either those two are multiple of two, or that one's a multiple of to This can't be a multiple of two.

We know P is not even because you can't have two is a factor.

So both of these numbers have tohave.

They're both multiple of two.

These are both even numbers, in fact, because they're two consecutive even numbers.

One of them, we don't know which one of them is a multiple of four.

So either this one's multiple of two.

This one's multiple of four or that one's for on that one's too right.

But we know when we multiply them together, the combined total will be a multiple of eight.

So now we know this.

It is a multiple off eight because it's too even numbers either side of a prime.

And once for once took, we have to get ites.

We've also got three numbers in a row and every batch of three numbers.

One of them has to be a multiple of three.

Every third number is a multiple of three again.

It's not the middle one, because you know that when there is not a motor pool off three because it kind of three is a factor, it's a prime.

So one of these has to be a multiple of three again.

We don't know which.

When we multiply them together, the total must be a multiple of three, so we also know that is definitely a multiple off three.

And if something is definitely a multiple of three, is definitely a multiple of eight, it is a multiple off 24.

And so that's it.

Just because the two numbers are either side of a prime.

If you multiply them together, you get a number, which is a multiple of 24.

And actually, we've not really used the fact that this is a prime.

All we've used is the fact that it's not even that is not a multiple of three.

So we've actually managed to prove is that all numbers, which don't have two or three is a factor.

If you square them, you get a number, which is one more than a multiple of 24 that's all the ones either side of every single six.

So what action should prove is both.

All primes are either side off multiple a six, and if you square any number on either side of a multiple of six, you always get a number, which is one more than a multiple of 24 that is one of my favorite prime patterns.

You said that your friends one there was easier.

It certainly is prettier.

Dilute.

It uses less, Inc but I'm not sure that would have been easier to have come up with.

That's very truth.

So I'm using easier, probably in a strictly mathematical sense, where I guess, easier.

I'm kinda using it to Maine.

This turning of the handle arguably, you're right.

This one is easier because there's not a lot of creativity.

I can say that it's my proof I can say it.

All I've done is just chunked into predictable categories and then turn the algebraic handle and tied it up.

And that's my response.

Where's this one?

Once you've got, it is easier to follow, but it wasn't easy to come up with, so I guess I'm saying it's easier from looking at it in hindsight, not easier coming up with it creatively in the first place.

But madman revelations love a creative proof, right?

And so the more creative you have to be coming up with the proof.

Normally, the more impressive mathematicians consider that proof.

I like to think you're on number five.

We go pretty deep into our topics, but I'm also aware.

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