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  • So I'm gonna be talking about graphs of sequences and I'm going to say graph.

  • I'm sorry I've lived in the US for so long.

  • I have to say graph.

  • Now, if you look at something like this, squares the squares Go one for now in 16 blah, blah, blah.

  • If you make a graph of them, this is a discreet sequence.

  • They separate numbers.

  • And I am as what started zero zero's term zero.

  • The one term is one the tooth, Miss four.

  • The third term is nine and so on.

  • So what we get is something that looks like that if we draw a continuous curve through it, If we plotted 1000 points, the dots would add merge together and we get something that looks like that This is a parabola single dots that it looks like a continuous curve Parabola.

  • You did it in high school.

  • Boring, boring, boring.

  • We're going to do more interesting things.

  • I'm going to show you some that you've never seen before.

  • But before I do that, here's another example.

  • The square root of an If we look at the square root of zero is 01 square root of two is 1.414 fruit to And of course, they're not whole numbers.

  • When I'm gonna plot in this picture is the nearest institute to the square root of end.

  • It's not a continuous curve anymore, because I'm rounding it off to the nearest institute.

  • So there are gaps like clusters are yeah, square root of 100 square, 101 squared of 102.

  • That's all gonna round off to 10.

  • So we get around 100 they're all 10.

  • So that's another example of a pretty pedestrian graph.

  • I want to show you some more interesting sequences.

  • Remember that movie called Avatar The Hero?

  • Jake Scully gets on the banshee for the first time.

  • He links to it and he tries to get the banshee to fly.

  • And it doesn't work.

  • Here's a sequence that reminds me of that.

  • I call this sequence fly straight.

  • Damn it.

  • And the definition is this.

  • We start off the 1st 2 terms of one.

  • Okay?

  • And now the rule kicks in.

  • What is the rule?

  • This is an and this is a van.

  • What's the next time you look at the last term and in and you compare them.

  • Are they relatively prime?

  • In this case, they are.

  • They don't have a common factor on what it is.

  • It's the previous term event, minus one plus n plus.

  • What So in this case, there's no common factor.

  • So the next term would be one plus two plus one, which is for what goes here.

  • Well, we looked at three and four.

  • Do they have a common factor?

  • They do not sow again.

  • The next term is for plus three plus one.

  • It's eight.

  • Now we look at this pair of terms.

  • What's the fourth term?

  • We look at foreign eight.

  • Ah ha.

  • They have a common factor of four.

  • When that happens, we divide the previous term by that common factor.

  • So here, the common factor this 48 divided by four is too so a event.

  • The rule is it's either the previous term plus n plus one.

  • If the greatest common divisor of a of N minus one and N is equal toe one, if they were relatively prime or it's equal to a of n minus one over G, C, D of n and A of n minus one.

  • If this G C D is bigger than one so that Let me let me see if I've understood it.

  • So there's a new common divisor there between five and to share.

  • Our next number is eight K.

  • Now there is a common divisor there.

  • Is it too?

  • Yes.

  • So eight divided by 24 Yes, No common divisor.

  • Next numbers 12.

  • Yes, Common divisor.

  • There is four.

  • Yes, The next numbers Three.

  • Yes, we have a common divisor again, which is three.

  • So it's one.

  • Yes.

  • Now we've got 10 and one so 12 there's no common divisor.

  • There's there's, like, 12 no common divisor there.

  • So no, we jump up to 2044 so on.

  • When we look at the graph of that, it's just wild.

  • It is totally random looking random array of dots until we get to term 600 something.

  • And at that point, Jake Sully says to the demon, Fly straight, damn it!

  • And from that point on, you remember in the movie, the music is chaotic, the banshee is tumbling, they're falling to the bottom of the cliff, whatever it iss.

  • But then suddenly the music calms down and they fly smoothly.

  • So What's happening is this.

  • Situation changes when we get to a of two.

  • M equals one.

  • Here was a of nine.

  • We got a one at the ninth stay.

  • It's not until we get to term number 600 whatever it is that we first get a one at an even term.

  • And when that happens, let me show you what happens.

  • We'll have an even numbered term, which is a one.

  • So there's in.

  • Then there's a event now what's the next term?

  • Well, it's gonna be termed number two M plus one.

  • What's its value?

  • There's no G c D.

  • Here it's one.

  • The next term is the sum of this, plus this plus one, which is to m plus three the next one.

  • It's too m plus to the G.

  • C.

  • D is obviously one.

  • There's no common factor there.

  • So again, we add when we get four m plus six term number two m plus three ah, to impress three foreign plus six g.

  • C.

  • D is to employ us three.

  • We divide to implicitly into that, and we get to good.

  • Okay, let's get the next four terms.

  • So to M plus three and next to him is gonna be two in place for the G C.

  • D of two.

  • And to him, plus war, it's too.

  • So we get to divide this two by two and we get one.

  • Okay, let's keep going and you'll see the pattern right away.

  • Now we get to M plus seven to him, plus six term again.

  • There's nothing.

  • So it's It's four m plus six plus seven plus one and then two.

  • M plus seven.

  • What is the G C D Bingo?

  • It's too m plus seven.

  • We divide that into that.

  • When we get to and then we continue.

  • What we get is one to M plus 11 for M plus.

  • This went up by eight.

  • It goes up by eight again in a two.

  • And so on.

  • What happened is we had four nice numbers in a row, which is what you see here.

  • There are actually two numbers together here and in that in that.

  • So with these numbers are on three straight lines.

  • So although I can't say then all that's a double line.

  • That's 212121 down there on that line.

  • Exactly.

  • Yes, yes.

  • What?

  • Those numbers are is we went from to M plus 3 to 2 in place 7 to 2 M plus 11.

  • We've added four and we've increased the value by force that slope one and this one, we've gone up by eight.

  • So it's slope too.

  • So this is slip one on that slope, too.

  • So from then on, the Banshee and Jake Sully fire fly smoothly.

  • It seems so arbitrary that we didn't hit that magic point until here.

  • Like it, you know, it was just It wouldn't drink the magic of mathematics.

  • Yes, it's just a strange fact about numbers.

  • More another graft.

  • Another amazing.

  • Another amazing craft.

  • The next graph we look at the primes, we don't, but we don't just plant the primes.

  • We write the primes in base, too.

  • Reverse them and subtract.

  • So let me do it.

  • The first prime is too.

  • Which in finery is one not and we reverse it and we get, nor one which is one we subtract and we get one.

  • So it's p minus.

  • Reverse pee in binary.

  • All right.

  • Next one is three.

  • Reverse it.

  • We get 11 11 from 113 minus three.

  • Is Syria so the next term is era 511110711 It'll get more interesting in a minute.

  • Zero Next prime is 11 binary.

  • That's 1011 When we reverse it, we get 1101 which is, you know, is 13 11 minus 13 is minus two.

  • So the final term that we spit out that second base 10.

  • Yes, The next prime Ms 13.

  • Well, that's just the other way around.

  • What we just did 13 is 1101 Reverse it.

  • We get 11.

  • And so they could sense to what makes to track.

  • The next one is 17 which is 10001 and we reverse it.

  • It's the same.

  • When we get zero 19 we get 10011 huh?

  • 11001 25.

  • So it's mine.

  • Six After 19.

  • What's the next prime?

  • 23.

  • When we reverse it, we get 11101 which is 29 23 minus 29 Seems to be minus six again and so on.

  • That's the sequence.

  • Take the finery.