ALEX CLARK: And what's
interesting about this problem is that it was actually solved
by what might be called an amateur mathematician, someone
that wasn't officially a mathematician. So I first encountered 163 when
studying a little bit of basic number theory. And it came up in the most
mysterious way that fascinated me because it's the
last number in a short list of numbers. Even though it seems very
innocuous, it has some very mysterious properties. So you may have heard about
factoring numbers. You may not remember exactly
what that means. So let's just review that. If you start out with a whole
number, say 12, you can write it as a product of numbers
in many different ways. So you could think of 12 as
being 6 times 2, or you could think of it as being
4 times 3. But if you want to write it as
a product of prime numbers, there's only one way to do it,
up to the ordering in which you write the numbers. So 2 and 3 are both
prime numbers. So every whole number can be
written in exactly one way as a product of prime numbers, up
to the order in which the numbers are written. BRADY HARAN: Any number? ALEX CLARK: Any number. BRADY HARAN: All right. I'm going to give
one, you ready? ALEX CLARK: No. OK. BRADY HARAN: 20. ALEX CLARK: That is going
to be 2 times 2 times 5. And each one of those
is a prime number. A mathematician named Gauss was
trying to identify all of the perfect Pythagorean
triples. So what are these? These are the whole numbers
a, b, and c that satisfy this equation. And what this means
geometrically, the reason these would be called
Pythagorean triples, is that these numbers are the side
lengths for a right triangle. The most common example of a
perfect Pythagorean triple would be 3, 4, and 5. And so the natural question
is, well, we can find specific examples. How do we find an exhaustive
list of all the examples of Pythagorean triples? And what Gauss noticed is that
well, if we take the idea of factorization of ordinary whole
numbers to a new number system, that we can make
some progress. And so the way that he created
this new number system was to take all of the whole numbers
and add i, sometimes denoted this way. So he takes all of the whole
numbers, that's what this zed corresponds to, and he adds i. So this consists of all of the
numbers that can be written this way, where the a and
the b are whole numbers. And i is the square
root of minus 1. Well, what he noticed is that
he could factorize this expression, if he just had i in
a way that can't be done in the ordinary number system. So if we take a squared plus b
squared, you might remember from school that this can't be
factorized in any nice way using ordinary numbers. If it were, instead, a
difference of squares, there would be a nice way to do it. But if we allow ourselves
to use i, we can actually write this. And what's more, the special
property that each number can be factorized into primes in
exactly one way carries over to this new number system. Now you have to expand your
notion of what a prime is because you've got
now new numbers. So they may or may
not be primes. And it could be that what used
to be primes are no longer primes in the new system. But what he also discovered was
that if we adjoin other numbers instead, we might
not get the same result. So for example, in this number
system, this is just all the normal numbers, except that we
now allow ourselves this one new number. So this one still is a number
system, but it has the unfortunate property that
it doesn't have unique factorization. So just to give you
an example. In this number system, we have
the number six, which could be written as a product
2 times 3. But, at the same time, it can
be written as this product. This is a problem because here
we have one way to decompose it into a product of primes, and
here we have another way that's completely
different, using completely different primes. It's not just a matter of
rearranging the ordering. BRADY HARAN: So it worked for
the square root of minus 1. It didn't work for the square
root of minus 5. Where are we going here? ALEX CLARK: So what he observed
was that it doesn't work for all numbers, but he did
identify quite a few for which it did work. And specifically, he identified
a list of numbers. We still do have the unique
factorization, provided we choose one of the following
numbers, d. Well, one-- these are the numbers for
which we get unique factorization. And then there's a big
jump to this number. BRADY HARAN: So that
one works. ALEX CLARK: And this
one works. As you can tell, it's quite
a bit removed from the previous one. And what Gauss conjectured was
that this was the last one. That for some reason, with this
number, the list stops. And no matter what other number
after this you add, you no longer have unique
factorization. He conjectured that this was
really the way it was, that it was true, but he could never
actually prove it. BRADY HARAN: I can see why that
number intrigued you now. ALEX CLARK: The problem is to
try to actually show that this is the last number for which
this happens, that this list is exhaustive. That there are no other numbers
of this sort that one can add in order to get unique
factorization in the new number system. BRADY HARAN: Tough nut to
crack then, was it? ALEX CLARK: It was a very
tough nut to crack. And what's interesting about
this problem is that it was actually solved by what might
be called an amateur mathematician, someone that
wasn't officially a mathematician. So a German by the name of
Heegner actually found a proof in the '50s. And he did actually publish
his result, but it wasn't accepted by the mathematical
community for several years. And then in the '60s, some
well-established mathematicians, a British
mathematician by the name of Baker and an American by the
name of Stark, found other proofs which were generally
accepted. And then, after a couple
of years, one of those mathematicians, Stark, went back
to what Heegner had done, analyzed it in more detail,
and realized, hold on a second, he did actually
have a proof. It was actually OK with just
some minor issues. BRADY HARAN: So, and what
did Heegner find? Was Gauss right? ALEX CLARK: He was right. This is the end of the list. And mysteriously, these
are the only ones. And there are some other
interesting and mysterious consequences of this that you
wouldn't guess just from the way these are presented. So one of these is the fact that
there are certain numbers that Ramanujan had identified
as being very close to whole numbers. So among these are this
number, this number. And who knows how he managed
to determine this. But Ramanujan did observe that
these were close to whole numbers, surprisingly close. And as a sort of joke in the
'70s, the recreational mathematician, Martin Gardner,
wrote in one of his columns that Ramanujan's conjecture
had finally been proven, namely that this was
a whole number. So here we've got
the number e. Here we've got the number pi. And this is taking
e to this power. So e is anything but
a whole number. It is as far from a whole number
as you could expect to be, likewise for pi. The square root of 163 is
also not a whole number. So it would be extremely
surprising if you took this combination and somehow wound
up with a whole number. And you don't. But he was able to fool people
because it's so close to a whole number that, with the
techniques that most people would have had available in
those days, they wouldn't have been able to tell
the difference. It's extremely close to a
massive whole number. Makes it very difficult to
check exactly what it is. But it turns out to be different
from it by just ever so slightly much. BRADY HARAN: Is the fact that e
to the power of root 163 pi is very close to a whole number
in any way related to all this stuff we've already
been talking? ALEX CLARK: Yes. In fact, the fact that you get
all of these from those is directly related to that fact,
not in an obvious way. It's quite involved to see
exactly how that works, but it is, in fact, related to that. And I'm not even sure how it is
that Ramanujan was able to identify these. I'm not sure that he knew about
these properties, but he did identify those as being very
close to whole numbers. Very strange. It's not something that I think
ordinary people would ever have expected. And who knows why it was that
Gauss conjectured that this would be true? Because one might also expect
that, since you have this big gap between 67 and 163, that
the next one might just be very far along. And before people had
computers, how would they have known? How could they possibly check? So I wouldn't have
guessed that. I don't know who would, other
than someone like Gauss. But it's a very mysterious fact,
but one that has now been established. BRADY HARAN: That's why
you think 163 is a pretty cool number? ALEX CLARK: It is a
very cool number. BRADY HARAN: Is it like your PIN
number or the combination on your briefcase? ALEX CLARK: No, no, but
it is certainly one of my favorite numbers.
163 is also responsible for the fact that n2 + n + 41 is surprisingly often a prime. Apparently.
As an amateur mathematician (I did not manage to retain much from class up to vector calculus), I find this extremely exciting and thoroughly enjoyable!
Are there other series available that "tackle" curious or otherwise interesting maths in a similar manner? I have seen a video about pegs and soap bubbles, and would be greatly amused by anything even tangentially related.
These rings are all examples of Dedekind domains.
There is some really beautiful geometry here. It turns out that we can think of a Dedekind domain as some kind of smooth curve C. If we zoom in closely enough on the curve, we always see the property of unique factorization, but it may not apply "globally".
For this reason, the question of whether some Dedekind domain is really a unique factorization domain turns out to be a very geometric one; it is the same as the question of whether C has no nontrivial line bundles.
You can think of a nontrivial line bundle as being a knotted thickening of the space—for example, the Möbius band gives a nontrivial line bundle over the circle. On the other hand, a straight line has no nontrivial line bundles, because any knotted strip of paper can be untied.*
So you can think of unique factorization as some kind of "straightness", and you can think of a nonunique factorization as some kind of knot or twist that can form in our mysterious curve C.
*I am simplifying somewhat, because there are other ways for bundles to be equivalent—but this gives the flavor.
This was titillating. But in today's age of computing, does 163 still reign supreme? Haven't we managed to crunch trillions on supercomputers to verify that?