 You know the tale about
Fermat's Last Theorem.
In 1637, Pierre
de Fermat claimed
to have the proof of
his famous conjecture.
But as the story
goes, it was too
large to write in the
margin of his book.
Yet even after
Andrew Wiles proof
more than 300 years
later, we are still
left wondering, what proof
did Fermat have in mind?
The mystery surrounding
Fermat's last theorem
may have to do with the way
we understand prime numbers.
You all know what
prime numbers are.
An integer greater than
one is called prime
if it has exactly two
factors, one and itself.
In other words, p is prime
if whenever you write p
as a product of two integers,
then one of those integers
turns out to be 1.
In fact, this definition works
for negative integers, too.
We simply incorporate
negative 1.
But the prime numbers
satisfy another definition
that maybe you
haven't thought about.
An integer p is
prime if whenever
p divides a product
of two integers,
then p divides exactly
one of those two integers.
Let's call this Definition
B. And let's think about it.
Does it sound plausible?
Here's an example.
Suppose our prime is 3.
And notice that 3
divides 12, for instance.
Now, look at the
different ways 12
can be factored as a
product of two numbers.
What do you see?
No matter how we
write 12, 3 always
divides one of the two factors.
You may think that's
a silly observation,
but it does not hold for
composite or non-prime numbers.
For example, 4 also divides 12.
But 4 does not divide 2,
nor does it divide six.
And the idea is that
this observation will
hold for all multiples of 3.
For example, 3 also divides 30.
And no matter how you write 30
as a product of two numbers,
3 will always divide
one of the factors.
Now, 6 also divides 30.
But it does not
have this property.
In particular, 6 is not prime.
So this new definition of
prime is perfectly valid,
even though it's not the
one that we're so used to.
So you might
wonder, why don't we
ever hear about Definition B?
Is it because these
two definitions
are actually conveying
the same concept?
In other words, is
every integer that's
prime in the sense of A also
prime in the sense of B?
And conversely, is every integer
that's prime in the sense of B
also prime in the sense of A?
It turns out the answer
is no, not always.
That is, the answer
is yes if you're
working with the integers.
In fact, I encourage you
to get out pen and paper,
pause the video and prove
that an integer satisfies
Definition A if and only if
it satisfies Definition B.
However-- and here's where
it gets interesting--
if we replace the integers
by a different number system,
a system where we can still add
and multiply and factor things
just like we do with integers
but where those things aren't
necessarily integers,
then it is not always true
that these two
definitions coincide.
To see why, let's
look at an example.
Let's replace the integers
by a different number system.
What exactly?
Well, in Gabe's episode
"Beyond the Golden Ratio,"
he explained how phi,
the golden ratio,
is just one of a family
of metallic means.
But the golden ratio also
lives in a different family.
Phi is the number 1/2 plus 1/2
times the square root of 5.
But what about other
numbers of the form
a fraction plus a fraction
times the square root of 5?
There are infinitely many
numbers of this form,
and the golden ratio
is just one of them.
The set of these
numbers form what's
called a quadratic field,
which plays an important role
in algebraic number theory.
But for the rest of the episode,
let's just focus on the case
when A and B are integers.
Collectively, we'll
denote these numbers
by z adjoin square root of 5.
The nice thing is that
we can add and multiply
these numbers together.
For example, to
add 1 plus 2 root 5
and negative 4
plus 3 root 5, just
add the integer parts and the
square root parts together.
So their sum is negative
3 plus 5 root 5.
And we can also
multiply them together.
We'll just use the familiar
distributive law, which
some folks like to call FOIL.
So their product is
26 minus 5 root 5.
Moreover, Z adjoined root 5
also has prime numbers given
by Definitions A and B.
But because we replaced
the integers with Z
adjoined root 5, we need to
modify the definition a little.
The reason is that Z adjoined
root 5 may contain numbers that
behave like the number 1, even
though they aren't the number
1.
I'll explain.
Here's the new Definition
A. A number p and Z adjoined
root 5 is prime if
whenever you write p
as a product of two numbers
then one of them is a unit.
A unit is a word that means
"has a multiplicative inverse."
That is, a number u is a unit if
there exists some other number
v so that u times v is 1.
For example, in the usual
integers 3 is not a unit.
It does not have a
multiplicative inverse.
OK, yes, 3 times
1/3 is equal to 1.
But 1/3 is not an integer,
so that doesn't count.
In fact, the only units
in Z are 1 and negative 1.
And that's why "unit" is a good
generalization of the number 1.
OK, so we have two
definitions, A and B.
If we work with the integers,
then these two definitions
coincide.
But now I claim that because
we are working in Z adjoined
root 5, they do not coincide.
In particular, the number
2 is prime by Definition A
but not prime by Definition B.
First, let's see why 2 is not
prime according to Definition
B. Notice that 4 can be
written as 2 times 2,
but it can also be written as
1 plus root 5 times negative 1
plus root 5.
This means that 2
divides the product.
But 2 does not divide
either factor, 1 plus root 5
or negative 1 plus root 5.
In other words--
and you can verify--
there are no integers a
and b so that 1 plus root 5
equals 2 times a plus b root 5.
Similarly if you
replace 1 by negative 1.
This shows that 2 is
not prime according
to Definition B. However, it
is prime by Definition A. Why?
I'll let you work that one out.
That's a little trickier,
but not too much.
I recommend using a
proof by contradiction
along with something
called a norm.
I won't go into the computations
now, but if you're interested,
check out the references below.
All right, let's summarize.
We have two
definitions, A and B.
When working with the
integers, these definitions
imply each other.
But in Z adjoined
root 5, they do not.
Why?
The reason is
because the integers
possess a very special
property that z
adjoined root 5 does not have.
Before I tell you what that
property is, let me just
say that this overall discussion
is a part of something
called ring theory,
the study of rings.
But not this kind of ring.
A ring is a mathematical
object, a set
of elements that behave
a lot like integers
even though they
may not be integers.
And Z adjoined root 5
is one such example.
The neat thing is that
once you have a ring,
you have enough
mathematical structure
to talk about primality.
In particular, our two
definitions, A and B,
have technical names
in ring theory.
An element and a ring
is called irreducible
if it satisfies Definition
A, and it's called prime
if it satisfies Definition B.
So earlier, we saw that 2 is
irreducible in Z adjoined root
5, but it is not prime.
Now, here's the punchline.
Primality and irreducibility
will coincide if and only
if your ring has a
very special property.
And the integers
have that property.
What is it?
The fundamental theorem
of arithmetic-- namely,
that every integer has
a unique factorization
into a product of primes.
More generally, if you're
looking for a buzzword
the integers form a unique
factorization domain, or UFD.
And according to
abstract ring theory,
irreducible and prime
are equivalent concepts
if and only if your
ring is a UFD--
specifically, if
each element can
be uniquely written as a
product of irreducible elements.
What's interesting is that
not all rings are UFDs.
And this brings us back
to Fermat's last theorem.
The absence of
unique factorization
is precisely why one
of the many attempts
to prove Fermat's Last
Theorem wasn't successful.
In 1847, French
mathematician Gabriel Lame
thought he had proof
Fermat's conjecture
by factoring an expression
like this, which
occurred in the ring Z
adjoined alpha, which is not
a unique factorization domain.
And so his technique
didn't work.
Fortunately, having a faulty
proof isn't always a bad thing.
In fact, the lack of
unique factorization
was spotted a few years
earlier in a different setting
by German mathematician
Eduard Kummer, who
introduced what he called
ideal numbers, precisely
to get around the issue.
In short, the discovery that not
all number systems, or rings,
have an analog to the
fundamental theorem
of arithmetic set
the stage for more
than a century's worth of
brand new mathematics, which
then led to Andrew
Wiles' proof of Fermat's
Last Theorem in 1993.
So what proof did
Fermat actually
have in mind when he
wrote in his margin?
Well, I'm not a
historian, but it's
very possible that he
assumed that properties
of the integers, like
unique factorization
and the equivalence between
prime and irreducible,
will always hold, just
like Lame thought.
But as we saw today, things
aren't always what they seem.
If you'd like to learn more
about the ideas discussed
in today's episode, be sure
to check out the links below.
See you next time.
 I'd like to start
by thanking everyone
who supports us on Patreon.
Amounts big and small help
keep the lights on here
at PBS Infinite Series.
We genuinely really
rely on your support
to help the program keep
going, and we'd particularly
like to thank Roman Pinchuk,
who is our first Converse Level
Supporter.
Extremely generous.
I don't know what
to say but thanks.
Now, let me get to
some of your comments
from our earlier episodes
on the Peano axioms
and the construction of the
natural numbers using sets.
First off, I want to shout out
Robert Lowe from the UK, who
we talked on Twitter.
I believe he's a viewer.
But he wrote a blog post
about very similar information
recently, the Peano
axioms and the formulation
of the natural numbers
in sets, and so forth.
He and I had a chat
about whether you
should start the
natural numbers at 0,
which is a big controversy.
Anyway, his blog is very good,
and the post is excellent.
It's a good supplementary
take on the material
that we did here.
The link to it is
in the description.
You should go check it out.
Gerard Tan made the
highly updated comment
that, for someone who was saying
let's not talk about numbers,
I was numbering the axioms
one, two, three, and so forth.
As many people pointed out
in their replies to Gerard,
those labels are arbitrary.
I could have called them cheese,
Cocoa Puffs, and Frankenstein.
Who cares?
But point taken.
Touche.
Jon Laing, or
"Laing," pointed out
the similarity between
the Peano axiom
formulation of the naturals
and the lambda calculus.
And other people also mentioned
explicitly the Church numerals.
The relationship between all
this stuff is no coincidence.
This is how the numbers
can be formulated
from an abstract computer
science perspective, as well.
I encourage you guys to Google
that stuff-- lambda calculus,
Church numerals.
On a somewhat related note,
Jesse Maes, or "Mays,"
pointed us toward the book
"Software Foundations"
by Pierce.
I checked it out.
Looks like it's a good resource.
I added a link to
the author's website
down in the description
of those earlier videos.
OK, a lot of you-- including
Gianluca Basso, EmNarr91,
and Joey Beauvais-Feishauer--
hope that's pronounced right--
brought up that there
are nonstandard models
of the natural numbers that
can be extracted from the Peano
axioms, at least in the
standard first order logic
formulation of the axioms.
But as Andrew
Kepert pointed out,
the version of the axiom
induction that I gave
is the second order
logic formulation
where you're quantifying over
sets and subsets and so forth.
So for people who don't
know the distinctions
between first and second
order and higher order logic--
I assume that's the majority
of the audience-- don't
worry about that conversation.
But for the rest
of you, the reason
that I tried to sidestep this
and why I made the disclaimer
that I made in the second
episode, which maybe I
should have made earlier
in the first episode,
was because I wanted to make
this as accessible as possible,
just to give people an
introduction to these ideas
without having to get
hung up on the nuances
of first order versus second
order logic formulations.
So those are the reasons
I made those choices
and tried my best.
mabamme, or "ma-bamay," asked
whether it's a good analogy
to think of the von Neumann
or Zermelo set theoretic
constructions as
an implementation--
this is a computer analogy--
and to think of the Peano axioms
themselves as just an interface
of the natural numbers.
I think that's a great analogy.
Craig Tyle pointed out that
in my little philosophical
aside at the end of
the second episode,
I probably should have reference
Paul Benaceraf, the philosopher
of math from Princeton
who championed
the structuralist
view of mathematics.
You're right.
And this just goes to show
how weak my philosophy
Fu has become that I'd
forgotten about Benaceaf.
I haven't read him
in over a decade.
But you're right.
I mean, these are not
original ideas on my part.
They've been championed
elsewhere in the literature.
Good call.
Finally, David
Gallo said that Gabe
should be given a million
bucks to do this YouTube
stuff full time.
I agree.
Now, I already get a
huge pile of cash, which
is why my clothes look so nice.
But another million
wouldn't hurt.
So Elon Musk, you watching?
Know what I'm saying?
