[0:00] I have a story to tell you about prime
[0:01] numbers which I think is pretty
[0:04] interesting and it's a new way of
[0:06] looking at primes. I it's ridiculous to
[0:08] say there's anything new about primes,
[0:10] but you'll see. I'm going to um start by
[0:14] remarking that primes are both very
[0:16] irregular and very irregular. They're
[0:18] irregular because when you look at where
[0:21] the primes are on the number line, they
[0:23] grow like weeds. Every so often there's
[0:26] a weed, a prime number, and then there's
[0:27] a gap and then there's another weed. On
[0:29] the other hand, if you look at the
[0:31] picture in the large between one and a
[0:34] million, say the mathematical notation
[0:38] for that is pi of n. Pi of a million is
[0:40] the number of primes between 1 and n.
[0:43] Look at the the primes between one and
[0:46] 50,000. How many are there? And it sort
[0:49] of looks like this.
[0:53] Rather surprisingly, when you get up to
[0:55] 50,000,
[0:58] it's basically a straight line. So,
[1:00] they're not that irregular. The famous
[1:03] mathematician said, "This is one of the
[1:06] most astonishing things in mathematics,
[1:08] the smoothness of this function pi of
[1:11] n." What it is, pi of n is very close to
[1:16] n / log n. And that really is
[1:18] astonishing when you consider how
[1:20] irregular they are at the beginning.
[1:22] This line, it's not straight. It's not
[1:24] n. It's n over log n. But log n changes
[1:27] very slowly when you So when you draw
[1:30] this, it looks pretty straight. There's
[1:32] a slight droopiness to it. Neil, does it
[1:34] astonish you? Because primes are so
[1:36] fundamental, right? They're these
[1:37] fundamental important things. And I know
[1:40] like they do seem a little random at the
[1:44] start, but wouldn't something so
[1:45] fundamental, wouldn't it make sense that
[1:47] it was really simple and elegant?
[1:52] >> Why don't I think it's remarkable? Why
[1:55] log of n of all things? You know, why
[1:58] not
[2:00] square root of n say? Why log? I mean,
[2:04] log is one of those transcendental
[2:08] functions. It's a surprise. It looks
[2:11] pretty linear. So I thought to myself,
[2:14] how linear is it? What if we really
[2:16] tried to put straight lines through the
[2:19] primes? So what I want to do is look at
[2:23] points where the xycoordinates
[2:26] the x coordinate is going to be um k
[2:31] and the y coordinate is going to be the
[2:33] kith prime prime k.
[2:39] So it's begin going to begin the first
[2:41] prime is two. So the first point on this
[2:44] graph is going to be 1 comma 2. The
[2:46] second prime is three. So it's going to
[2:49] be 2 comma 3 and then the third prime is
[2:51] five and we get 3a 5 and so on. I want
[2:54] to look at those points and see how
[2:56] linear they are. All right. So I've made
[2:59] a piece of graph paper here going 0 1 2
[3:02] 3 and so on. And here's the prime. So
[3:06] the first prime is two. So that means we
[3:08] put a point at the coordinates 1 comma
[3:12] 2. That's called a prime point. The
[3:15] second prime is three and it's the point
[3:18] 2a 3. Then five then 7 then 11 then 13
[3:24] 17
[3:26] 19
[3:28] and then the next one is 23 and then 29
[3:31] and so on. There are the prime points
[3:33] and what I want to know is how close are
[3:36] they to being on straight lines. I'm
[3:38] going to define a sequence which is the
[3:40] number of lines we need to cover the
[3:43] first n primes with the smallest number
[3:45] of lines. So let me just do it and
[3:47] you'll see. So what's what's a of one?
[3:49] How many lines do we need to cover the
[3:51] first prime? We need one line. It can be
[3:53] anywhere you want. Just put it through
[3:54] that. So if n is one, the number of
[3:57] lines is one. What if n is two? We want
[3:59] to have a line through the first two
[4:01] primes. All right. I'll put a line. It
[4:03] goes through that point and that point
[4:04] as a straight line. One line is enough.
[4:07] two primes. I need one line. What about
[4:09] three primes? Dot dot dot. I want to
[4:12] have a line. There's no line that goes
[4:14] through those three points. Obviously, I
[4:17] need two lines to cover the three
[4:19] points. What about uh four primes? 357.
[4:22] 357 are in a straight line. So, I can do
[4:25] four primes with two lines. What about
[4:28] five primes? I can do with two lines. I
[4:30] use one line to cover three, five, and
[4:33] seven. and another line to cover two and
[4:38] 11.
[4:38] >> Oh, that the line doesn't don't have to
[4:40] touch each other, right?
[4:41] >> They can touch each other.
[4:43] >> They don't have to join. It doesn't have
[4:44] to.
[4:45] >> They don't have to connect. No, they're
[4:46] line segments. You just put them down.
[4:48] And of course, there could be primes way
[4:50] out here on the on the line. We'll worry
[4:53] about that later. I'm just trying now to
[4:55] cover the primes with as few lines as
[4:58] property. I'm I'm getting lines of
[4:59] primes. How many do we need? How many
[5:02] lines do we need? All right. So I did
[5:04] with uh with five. 1 2 3 4 5. What if we
[5:06] have one more? The line through those
[5:08] two does not go through that one. So I
[5:11] think we're going to need another line
[5:12] for six primes. We need three lines
[5:15] >> as we get higher and higher. This line
[5:17] here you used for example to join um
[5:20] three, five, and seven. Later on
[5:24] >> there may be a better way to do it.
[5:25] >> You might not that line might not exist.
[5:27] >> That's right. Yeah. Yeah. I mean the
[5:28] line exists but we don't need to use it.
[5:30] We're looking for the best way to cover
[5:33] the primes with straight lines. This is
[5:35] actually a famous uh an example of a
[5:38] famous computer science problem which is
[5:39] called set cover. You you you want to
[5:42] cover you you've got your set of objects
[5:45] and you've got a set of things you can
[5:47] use to cover them and you want to find
[5:48] the optimal the smallest subset of your
[5:52] objects to cover the things you're
[5:54] trying to cover. Here we're trying to
[5:56] cover the primes and we're using lines.
[5:59] Is this like a mathematical thing or is
[6:00] this a game? Like
[6:02] >> it's a mathematical thing. It's a deadly
[6:04] serious thing. Oh yes.
[6:06] >> Deadly serious.
[6:07] >> No. Well, not deadly serious. No,
[6:08] there's no money at stake. There's no no
[6:10] lives at stake.
[6:11] >> But it's a re It's a legit It's real.
[6:13] It's It's hardcore math.
[6:14] >> It's hardcore and it's new. The first
[6:16] time we need three lines is for the for
[6:19] six primes. When we get to seven primes,
[6:23] we want to cover 1 2 3 4 5 six seven. We
[6:25] want to go all the way out to here. And
[6:26] we can do it if we're clever with three
[6:29] lines. We can repeat lines through a
[6:32] point. And that and that is no. So seven
[6:34] primes we can do with three lines.
[6:37] >> Apparently you can look it up in this
[6:38] thing called the online encyclopedia.
[6:40] >> You certainly can. Yeah, it's sequence A
[6:44] 373813. You can look it up. So as I
[6:46] said, this is something that computer
[6:49] science people will say, "Ah, set cover.
[6:52] I can do that." It's it's one of those
[6:54] NPcomplete problems. So, it's not going
[6:56] to be easy to solve, but they do have
[6:59] good computer programs for attacking it.
[7:12] And my friend Ma Max Alex ran his
[7:17] program ran set a good set cover program
[7:21] up to 410
[7:23] primes. It looks like this. And more
[7:26] precisely, here it is. And you can see
[7:31] it's increasing and it's a bit
[7:33] irregular. There are long flat
[7:35] stretches. And you get a long flat
[7:37] stretch when you get a really good line
[7:40] that has a lot of primes on it. You
[7:42] don't you can just as we saw here, you
[7:44] don't have to increase.
[7:46] >> Or do they have names? Are they like
[7:47] called golden lines or
[7:49] >> No. No, they don't.
[7:50] >> They're like a seam of gold. They they
[7:52] are like a seam.
[7:55] A quick footnote. Since we filmed this,
[7:57] Max Alex save has calculated the lines
[8:00] required all the way out to the 861st
[8:03] prime. The plot looks like this. And
[8:05] there are two really interesting golden
[8:07] lines here and here. There are 48
[8:13] consecutive primes that can be covered
[8:15] by 68 lines. Then a whopping streak of
[8:19] 112 primes can be covered by 69 lines.
[8:24] But after that, nothing Max has found
[8:26] has come even close.
[8:29] And there's no reason to be found for
[8:32] this golden sweet spot
[8:36] that the sequence increases. It looks to
[8:40] me like it's roughly linear. I suspect
[8:42] it's actually about going like x over
[8:45] log x because that's in the nature of
[8:47] the game. But there are long stretches
[8:49] where you get a really good line that
[8:52] covers a lot of primes. And so you don't
[8:54] have to increase the number of lines
[8:56] until you get to the next awkward prime.
[8:59] And it looks like that. So
[9:02] >> that's a good name. An awkward prime. An
[9:04] awkward.
[9:05] >> An awkward prime is a prime that causes
[9:07] a step up.
[9:08] >> A step up. Yes. They're the awkward
[9:10] primes. Yes. Good. Yes.
[9:12] >> Can we can we have that name?
[9:13] >> All right. Sure. And the these
[9:15] >> Can you put is Can you put that in the
[9:17] OEIS?
[9:17] >> I think. Yeah.
[9:18] >> And will you call them awkward primes?
[9:20] >> Yeah. Sure. Sure. Okay.
[9:22] >> That's on tape now. That's on tape.
[9:25] >> An awkward prime causes a step up in the
[9:28] number of lines needed in your little
[9:30] line game here.
[9:32] >> Well, another footnote. You just
[9:34] witnessed the birth of the awkward
[9:36] primes. Here they are highlighted on the
[9:39] original sequence. there in the blue.
[9:44] And here it is, perhaps the most awkward
[9:46] prime of them all. The one that ends
[9:47] that whopping streak of 112 primes in a
[9:51] row that can be covered by 69 lines. The
[9:54] party pooper prime. And here they are,
[9:57] their very own entry in the online
[9:58] encyclopedia of integer sequences.
[10:01] Amazing. I'm a very proud co-father.
[10:07] >> Got so many questions coming into my
[10:09] head.
[10:10] >> Yeah. Yeah. Like what's the longest line
[10:12] for each number of primes? What's the
[10:14] longest line that's got three primes on
[10:16] it and four primes on it? And
[10:17] >> I'm coming to that.
[10:20] >> Like that Robert Graves poem about the
[10:23] Welsh the creatures that came out of the
[10:25] sea in Harl.
[10:29] >> That's a different story.
[10:30] >> Okay.
[10:32] But the last line is, "Ah, but I was
[10:34] coming to that."
[10:38] >> Puzzle alert, people. Puzzle alert. The
[10:42] diabolical geniuses over at Jane Street
[10:44] have cooked this one up to test your
[10:46] number skills. This is what it looks
[10:48] like. There are more details over on
[10:51] their site and via the links below. It
[10:53] is a doozy. For those who don't know,
[10:55] and there can't be many number file
[10:57] viewers who don't, Jane Street is our
[10:59] channel sponsor. They're a quantitative
[11:01] trading firm with offices in New York,
[11:03] London, Hong Kong, Amsterdam, Singapore.
[11:06] They use techniques from machine
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[11:14] when they aren't doing that, well, they
[11:17] don't mind a puzzle or two. In fact,
[11:19] there's a whole puzzle page on their
[11:21] site, which I've linked to down below.
[11:23] And while you're there, check out all
[11:25] the open rolls at Jane Street they're
[11:26] currently hiring for. Even though
[11:28] they're a financial firm, they don't
[11:30] expect you to have a background in
[11:32] finance or any specific field really.
[11:34] They're just looking for smart people
[11:36] who enjoy solving interesting problems.
[11:39] Now, why don't you go try those puzzle
[11:41] links?
[11:45] It's not just it's the concatenation,
[11:48] the stringing together of all the
[11:50] chunks. This whole thing is the
[11:53] sequence. And what we're drawing
[11:55] actually is a pin plot officially.
[11:58] There's one other sequence I've seen in
[12:00] the past. This was Yan Ritz Van X
[12:04] sequence that I did a video for you
[12:06] about. The Van X, the famous Van X
[12:10] sequence.