-: My name's Andrew.
I'm not a scientist, I'm a mathematician.
And usually when I say that I study math
that gets kind of the same reaction
as if I just said that I punched your grandma.
(audience laughs) So if that's you,
if you're math phobic,
then you know, no worries, tune me out.
It's a ten minute talk, just wait for the next guy.
But if you've got a little bit more patience then
you know I don't do science,
even my math doesn't really have any applications.
How exactly do I make this career out of
something that people just hate so bitterly.
Right, like what do I even see in it?
And I can show you.
So back in undergrad when I was a young warthog
you know I thought that I wanted, I got really interested
in this field of math that's called number theory,
which is the study of positive whole numbers.
So that's, from now on, all my numbers
are positive and whole.
The positive whole numbers,
especially the prime numbers.
But I went to this small liberal arts school
and it didn't have a class in number theory
so you know I thought, "Aw, man, no problem."
I went to the library and I dug up
this number theory textbook, went off to the second floor,
sat down and cracked it open to page one.
Right, 'cause optimal learning strategy,
sit by yourself, talk to no one.
(audience laughs) Write that down, it's a tip.
But you know, despite that right,
the first few pages went pretty well.
It was stuff that you know I had sort of
picked up along the way as a math major.
Right, like I know what a prime is.
I know how many primes there are.
I know what we do with primes.
Uh-oh.
Oh no.
Okay, let's try this.
Yeah, that's what I meant.
There we go.
So you know, none of that stuff was new to me.
But I only made it to page five
before I absolutely just ran headlong
into this crazy wonderful beautiful problem
that I had no idea, never heard of before
and before I share it to you
kinda needs a little backup, or background.
The primes are absolutely just totally a mystery.
We just don't know.
There's some that we can say,
especially just like global big large scale
patterns that we do understand,
but there's a lot of things that we don't.
For example, like this handy dandy Happy Birthday balloon.
If I hold it like this and take off this orange clip,
I can tell you that the air is gonna go that way.
But there air in this balloon is bouncing around
at you know 70 degrees Fahrenheit, right?
If you asked me, "Andrew, what's going to happen
to this particular air molecule right here?
Which direction is it heading?"
Pfft, I don't know.
I don't know, they're bouncing around, right?
I can tell you that on average
the air is going that way.
I cannot tell you where any molecule is going.
And the primes are the same way,
that I can say big level things but I
absolutely can't tell you small level.
So here's a big level pattern.
The primes get rarer as they get larger.
Out of the first hundred primes,
the first hundred numbers, there's 25 primes.
Out of this hundred numbers, there's only 12.
But despite that, right, this is a true fact,
the primes do get rarer as they get larger.
If you hand me a big number it's a very
small probability that it's going to be prime.
And the gaps between them grow, and grow, and grow,
except sometimes not.
Those are as close as two prime numbers can get, right?
You can't be even and a prime
unless you're specifically two.
So primes that differ by two like that are called twins.
And twin primes appear to persist.
There are small ones, there are medium ones,
there are big ones,
and there are many, many, many more.
Which is like nuts, right?
How can, I literally just said that
they get rarer, except no, they don't?
They're close together, and far apart?
Like, how, what?
What does that even mean?
But that's also kind of not a surprise
with this physics metaphor that I'm gonna make.
The primes, the individual ones
are sort of pseudo randomly distributed right,
you know, we can't predict them.
So I have these lima beans
(beans hit the floor)
and now they're pseudo randomly distributed on the ground.
(audience laughs)
Some of them scattered, right?
But some of them are close together.
It's not a surprise that some of the beans piled up.
And so it's not a surprise that some
of the primes are only twins,
only two away from each other.
And this was first formally stated
as the Twin Prime Conjecture in the mid 1800's,
that the available evidence seems to suggest
that that never stops happening,
that there are infinitely many such twin primes.
That's what I found on page five.
It's so crazy, it's so cool but it was,
you know everybody seemed to suggest you know,
we'd find these primes and we'd keep finding them.
It appears that it's true but nobody could prove it.
It just sat around for over 150 years
until this guy named Zhang in 2013
absolutely came out of nowhere,
hadn't published a paper in like a decade.
He proved that there are infinitely many
primes that different by N,
where N is some number than he couldn't pin down
but is was less than 70 million.
(audience laughs)
It was like a joke, on yeah, oh big help, right?
But he proved that theorems in this area are possible,
like this is an attackable area and I can
put a finite bound on it.
And that was enough to get him a MacArthur Genius Grant,
if you've heard of those.
And it absolutely set the research community on fire.
There's this huge collaborative team
of mathematicians that within a year,
they'd improved and found refinements
in his arguments, that kind of stuff.
There are infinitely many primes that differ by N.
They got the 70 million down to 246.
And then there's this other related
conjecture in the same area that
if we add that as one of our hypotheses,
you can get it down to six.
And if you would like to become world-famous
and get a tenure track job at
a university of your choice,
replace the six by a two.
But despite all this, this you know just
amazing, wonderful, crazy problem,
this book that I was reading casually mentioned
in less than a single sentence that obviously
triplet primes can't happen.
There's you know three and five, that's a twin,
but three, five, seven, those are triplets.
Eleven, 13 that's a twin, but 11, 13, 15,
not a triplet, 'cause 15 factors.
What? That killed me.
How can like, twin primes are just so impossibly difficult
that if you make 70 million degrees of progress
you get a MacArthur Genius Grant,
and triplet primes are just so easy that
he doesn't even work out the argument,
he just says, "Oh yeah, sure, clear."
How does that even happen?
So I shut the book.
I had to do a dot dot dot 'cause if
talking to a scientifically literate audience
would kind of give it away.
But I shut the book and I tried to figure out
what was that reason, what happened?
But how do, if you're not familiar with
mathematical style argument,
what am a supposed to do to talk to you about prime numbers?
Am I, you know take a hundred primes
and give 50 of them a placebo?
What, how do I even take a sample of prime numbers?
They're are infinitely many of them,
how can I even put a dent in infinity?
The idea of experimentation literally
doesn't even apply to this question.
But none of that dumb stuff is going to stop me.
(audience laughs)
And we can do it right now.
So here's something that we can say,
despite the fact they're we're having
to stare down infinity and we can't take a sample.
Here's a true fact.
If you divide any positive whole number by three,
you either get remainder zero,
remainder one, or remainder two.
You can't get remainder three because
that means that you're not done dividing, right?
And here's the idea I had,
sitting at this graffiti-covered desk,
staring at the back of the theater building.
Is that I can rephrase this using algebra.
The number that leaves remainder zero when
you divide by three is a multiple of three.
It's three times something.
And like we usually do, I've chosen to
represent "something" with the letter X.
And that means that if you leave remainder one,
you're three X plus one.
And if you leave remainder two, you're three X plus two.
And once we have this the variable defined in hand
and we can say every number fits in exactly
one of these buckets,
then it's really easy to see why there's no triplets.
One of them has to be a multiple of three.
Let me show you right now.
So it's possible that our first triplet
is a multiple of three itself right,
that's one of our choices?
In which case, that's the form that
the other two triplets take.
But who cares, 'cause the first one is already not prime.
It's also a possibility that our second triplet
leaves remainder one.
In which case, look at the other two triplets,
specifically the middle one.
I can factor a three out of that.
Or as I like to call it, not prime.
Also it's a possibility that our first triple
leaves remainder two.
In which case, check out the last triplet.
I can factor a three out of that one, which makes it-
-: [Audience Members] Not prime.
Andrew: Not prime.
And that's it, we're done.
Every single possible number ever
fits in exactly one of those categories,
and every one of those categories is a problem.
There are no triplet primes except for, you know the dumb
trivial starting one, three, five, seven.
We did it.
I felt just so well, pleased with myself,
(audience laughs) shouldn't put a you know,
good veneer on it, but I just felt so you know,
this massive historical problem that was so
difficult and had all these books written about it right,
I'd taken it and brought it to my own doorstep
and I'd done part of it myself.
And now we've done it.
And if you personally, each one of you,
if you want to do something just
startlingly beautiful in math,
you don't need a lab, you don't need pipettes,
you don't need equipment, you need no machines,
you don't need an IRB, nothing.
(audience laughs)
All you need is pen, paper, and an idea,
and the pen and paper are optional.
So let me leave you with some questions.
I just told you that you can do math,
so here's some math to do.
What happened to one, three, five?
I said three, five, seven was the only triplet.
Also another question, are triplets like
the 11, 13, 15's of the world?
What about the 11, 13, 17's?
Any obviously reasons why, or for or against those?
Also one final question,
what happens on page six?
(audience laughs)
(applause)