---
title: 'Awkward Primes - Numberphile'
source: 'https://youtube.com/watch?v=VFoIPlUalRY'
video_id: 'VFoIPlUalRY'
date: 2026-06-28
duration_sec: 733
---

# Awkward Primes - Numberphile

> Source: [Awkward Primes - Numberphile](https://youtube.com/watch?v=VFoIPlUalRY)

## Summary

The video explores a new way to look at prime numbers by plotting them as points and asking how many straight lines are needed to cover them. This leads to a combinatorial problem related to set cover, revealing surprising patterns and 'golden lines' that cover many primes. The concept of 'awkward primes' is introduced for those that cause the number of lines to increase.

### Key Points

- **Duality of Primes** [0:00] — Primes are irregular locally but smooth globally, approximated by n/log n.
- **Line Game Definition** [2:56] — Plot primes as points (k, p_k) and find the minimum number of straight lines to cover the first n primes.
- **Set Cover Connection** [5:35] — This is an instance of the set cover problem, which is NP-complete.
- **Computational Results** [7:12] — Max Alex saved computed up to 861 primes, showing long flat stretches where few lines cover many primes.
- **Golden Lines** [8:19] — A streak of 112 primes covered by 69 lines is the longest found.
- **Awkward Primes** [9:07] — Primes that cause a step up in the number of lines are called 'awkward primes'.

## Transcript

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