Oh my goodness - we've got a fresh bit of
maths research, well from 2019, I consider
that
very recent all things considered; about
Gaussian integers, rings, finite fields,
and the magic square of squares. And so
for some of you that's a bit of a
spoiler on where parts of this video may
be going. Some of you may know the Parker
square - this is- a fan sent this in. If you
don't know what the Parker square is
that's great to hear. I don't want to
call it the Parker square because it
doesn't work properly- it would be like
oh that's a
classic Parker square. We'll get to this
content in a moment. First of all I
really want to focus on finite fields. So
a field is a bit it's like a generalised
version of what is a number; in terms of
you can multiply it, you can add it, and
very importantly you can find its
inverse. So for example the numbers, the
real numbers, are a field because if
you've got any- if you've got a number I
don't know a and you multiply it by b
you're going to get exactly the same
thing as if you've got b multiplied by a,
which means that they are commutative. And
there will exist some number, the inverse
of a. So if you multiply a by 1 over a
then you get 1. So that's our
multiplication inverse, and so for any
number - if you had for example a 4
and then you multiply that by 0.25
then you get 1. And very importantly
this is a real number,
this is a real number; and so every real
number has a buddy
real number where if they multiply
together you get 1 which means you can
undo multiplication, it means you've got
division
and you're off and racing. And you can do
this for all sorts of cool things, you
can do it- like I said for the real
numbers, you can do it for complex
numbers, you can do like some
polynomials if you want to go wild,
rationals - the field is wide open is what I'm
trying to say. Except
the integers.
The integers don't work.
Very sadly, big fan of whole numbers.
Everything else is great, you can add
them, you can multiply them - and if you
multiply them, this is the other thing I
should say, if you multiply two of them
together a times b
and you get some number c
that is also a real number - or it's it's
a member of the field. So the field's
closed, there's nothing you can do in the
field
that will cause the ball
to land outside the field. Anything- if
you're going to multiply or add you
land somewhere else in the same field,
very important and you've got this here.
Whereas integers, that works, you multiply
two integers together
you land somewhere else in the field of
integers;
but you cannot find
an inverse, a whole number inverse, for an
integer so that means they get kicked
out.
Unless you
shrink the field a bit. So instead of
having the infinite-
as I say there's infinitely many
integers, there's a big infinity of real
numbers. Instead of having
too much space, we want to we want to
narrow this down a little bit, if you
have a finite field
suddenly your integers are back on the
table
and you can have inverses of whole
numbers
provided the field is finite. You know
what, let's very quickly put together
the addition and multiplication grids
for a finite field across some integers.
So Brady, do you have a fun
prime number? And I should warn you the
bigger the prime you name the longer
this is going to take. Don't go trivial!
Don't- - (Brady: 7.)
7, that's nice. That-
that's neat and tidy. So what we'll do is,
so for 7 we're gonna effectively be
doing
modular arithmetic on mod 7. And so what
we're going to do is- let's do
multiplication first. And I'll very
quickly fill in this table. So we're
going to multiply,
here we go,
we're going to multiply
as if this is a grid so 1 times 1 is
1 - this is a super boring column.
This one's not much more exciting so 2
times 1 is 2, two 2s are 4,
6 - now this this should be 8,
mod seven - 1.
And so here we should have 10, so that's
going to be 3. And here we should
have 12 so it's going to be 5. So
I'm taking whatever you get when you
multiply these and I'm getting the remainder
mod 7. So I'm dividing out any 7s,
it's just the leftover bit.
And we can do the same thing for
addition, but we add them and we take the
mod 7 and we loop back. Now at this
point a lot of people thinking, well hang
on, why are you getting so excited about
finite fields when you're just doing
modular arithmetic? That doesn't count.
Well, it's not 100% true. So I would call
this- this is like the the integers-
so Z,
weird z for the integers, kind of mod
7 gives us this.
This doesn't work for any number, you
don't get a finite field-
which is why I said prime. It only works
for
primes and powers of primes.
So
you will get a finite field for the
integers mod 7, you'll get a finite field
for the integers mod 49, because that's 7
squared, or mod 25 because that's 5
squared or whatever the case. It doesn't
work for the in-between values. Now I've
never tried this
but I thought it might be fun
just to see what happens when it breaks,
so let's break it. Let's- Brady you want
to pick a number which is not a prime
and is not
a power of a prime
and then we'll we'll try to do the same
grid and we'll see we'll see why it
breaks.
(6?) - 6, excellent, because that's 3 times 2
it's not a power of a prime, it's got
more than one prime factor is the other
way of saying that. Ao so this is going
to be fancy Z6.
Okay so I should point out I- I've kind
of hidden 0, because a whole row of
0s is even less exciting than
having these.
And 0 doesn't get inverses and all
that jazz so we ignore 0. Now, so you
think well this is perfectly good, why
doesn't this count as a finite field?
Well it's because we've haven't got our
inverses. So if you ever look over here
all the numbers we might have, all the
non-zeroes, 1 through 6; 1 has an
inverse, it's 1 because there's 1
here. 2 has an inverse,
it's 4. So if you multiply 2 by 4
mod 7
you get back to 1. So technically a
half
in
the finite field
7
is 4.
So
that checks out. Three- a third is 5
because 3 times 5 gives you 1 and then
there's 1 there, and there's 1 there
and there's 1 there. And so quite
nicely there is a 1 in every single row
and there's a 1 in every single column.
(As you said up here, we have to be able to)
(achieve 1 somehow?) - Yes you have to
multiply to get to 1; and you can't do
that with regular integers but because
you've got mod it means you can loop
around to get back to 1. Because
otherwise once you- once you're gone
you're not you're not coming back to 1,
whereas now
loops back. However,
sometimes,
you never hit the 1. So there's a-
there's some 1s in this grid but look
at this, 2 hasn't got a inverse. There's
nothing you can multiply 2 by to get
back to 1 mod 6, and so that's why this
one is ruled out. It's still nice, still
modular arithmetic,
big fan,
it's not a finite field. And finite
fields then linked to all other amazing
things, in loads of abstract mathematics
you'll be looking at
fields and finite fields which are great.
And finite fields have allowed for a
breakthrough in an area of maths very
close to my heart. When I first read a
paper I'm just kind of
glancing through it, I'm not checking
through all the details, I just want to
get a sense of what they're doing and
you know, what I do and don't understand.
And then what really jumped out at me is
section five here: 'The smallest
non-Parker finite field'. It turns out,
they're now using my name,
that's me I'm Parker,
as a type- like a a property that finite
fields can have. In fact you can put all
finite fields into two categories: you
can put them into the Parker finite
fields and you can put them into the
non-Parker finite fields. And a very good
question might be,
which is better?
And very sadly
a
a Parker finite field means it doesn't
work - it's like a dud. It's the dud.
Finite fields, when they're growing up,
what do you want to be -
when you're exactly the same size
because you're finite - and they say,
I hope I'm non-Parker. That would be the
dream, to be a non-Parker finite field. So
I've got a thing named after me-
(I'm afraid your field has been diagnosed as Parker-)
I'm afraid it's Parker.
It's not curable, the condition of being Parker.
Oh and they've used fancy F to mean
finite field mod 29, for example, is not
Parker
since:
and then they've given a magic square. So
for everyone who's unfamiliar with
magic squares of squares, this is a
really nice entry point. So this is a
magic square
and it's got nine numbers in it. And so a
magic square,
if you want it to be properly magic, is
if you add together all the columns you
always get the same total and if you add
together all the rows you always get the
same total.
And-
people are really insistent about this
it turns out - if you add together both of
the diagonals
you get the same total. And then you
can go insane; you can do all these like
pan diagonals that wrap around but
people don't tend to put those in. They
just care about both of those diagonals.
That doesn't work for these numbers,
until,
we square them. You now have a magic
square, where every number in it is a
square number and it works. - (Hence magic)
(square of squares.) - Hence magic square of
squares. Now,
this has one extra condition so- so the
full backstory is I came up with a magic
square of squares because I thought it
was interesting, no one had ever found
one and no one had ever managed to prove
that one doesn't exist, which is still
true for the record. So I gave it a go. I
found this one. Now mine has the extra
condition it's not very good; and that's
because it's got duplicate numbers,
so you can- in fact it's symmetric which
is terrible. And it doesn't add- it
doesn't do all of them, one of the
diagonals is wrong, near miss. And this
has been dubbed the Parker square and I don't
know who was responsible for this, but a
line of merch came out.
People started showing up to my shows
wearing t-shirts - very few performers
have to put up with being bullied by their
merch at their own gigs. This one has a
condition as well.
So it is- it's got both diagonals it's
got every row, every column, except the
whole thing
only works mod 29.
So this works on the finite field of
integers mod 29.
We can- we can check that. I would like
to do that and you can decide how much
stays in the edit.
And now these are all mod 29, so anything
smaller than 29 doesn't budge; anything
bigger than 29 - woo!
So now
we need to go through and add these up.
So we'll do the rows first shall we? So
23 and 25 is 28, 29 - so if we come across
there, add them up, we get 29. This one
here, we get 29. This one here you're
thinking - whoa!
It's gonna be bigger than 29;
28 and 24 is gonna be 52,
58,
and you're like, that's way too big. Well
actually, 58
mod 29,
well it's zero. In fact all of these mod
29 are zero. So actually if you add every
single direction you get zero,
so it's a magic square of squares
where the magic sum is zero
so-
(That's beautiful.) - I don't know why I
find that so pleasing. - (And look what)
(dross you served us up!) - Hey!
I was four years early. Like you can't-
look like it's like looking at the
iphone 4 and going, well that wasn't very
good. Yeah, things have moved on,
technology has advanced. Without this,
this wouldn't be happening. - (That's right.)
What a weird world we live in Brady.
(So 29,)
(like that's a Par- that's a)
(non-Parker field.) - It's a non-Parker field
because
within
the numbers
in
the finite field of order 29
there exists a perfect 3x3
magic square of squares.
And so any finite field where you can
make a magic square
of squares
is a non-Parker finite field, because it
works. And then the rest of the paper- of
this section of the paper, this is really
quite nice, they then go they go well
hang on okay we found this one, and then
they say here's all the finite fields
with orders less than 29. And so you've
got a whole bunch of them here, they're
all the primes and powers of primes
below 29. So now they found the one for
29, and the obvious question is, is that
the smallest? This is what you always ask
in maths, is it the smallest? Is it the
biggest? And so
their first question was,
is 29 the smallest finite field which is
non-Parker? So is everything below that
unfortunately resigned to being Parker?
(And when you say finite fields, we're)
(narrowing this search to the integer)
(finite fields, obviously?) - Yeah, yeah. And
you could do magic squares with other
things, chuck some polynomials in there.
You could do complex- in fact Gaussian
numbers they're talking about earlier,
when when you're looking at it the
complex space. So there's a lot of
fun you can have. So yeah, they go
through, they rule them all out,
none of them are non-Parker. Everything
below 29 is Parker; and so they managed
to show
comprehensively that this
is the smallest non-Parker
finite field. But then they turn and look
in the other direction.
In general, they suddenly realise,
actually,
most finite fields
are non-Parker.
Because the Parker ones, the ones that
don't work, are quite rare. And so here
it's a conjecture, so this is not proven,
but they seem pretty confident:
'The only Parker fields of prime order
are these ones'.
And so it turns out any finite field
with a prime order of our friends the
integers, bigger than 67,
is non-Parker. All of them; the infinitely
many prime numbers bigger than 67, all
have a magic square of squares
in there somewhere in their finite field.
So
it turns out not working, being a Parker,
is actually quite a rare, special,
property. - (Prestigious.)
Exactly. Bestowed upon very few
people and finite fields
get to have the luxury of being Parker
because- and it's conjectured but I- I'm I
I'm going to believe them. Because once
you go past 67 they're all- they're all
non-Parker. Now, the flip side to this is
you might say, well what about
infinite fields? Like what what about the
integers which is what started this
whole thing in the first place?
For the infinite field
of
integers, well this is no longer
a field, it's now- it's a ring at this point.
It's- it's no longer a field, there's
infinitely many of them. The question I
have is does this tell us anything about
finding the Parker square in general?
Or- the true the true, yeah. And
sadly we still don't know.
So all they can say is
you can have arbitrarily large
finite fields
of a prime order and they will always
have a magic square of squares in there,
they're all non-Parker because they work.
We still don't know
if a magic square of squares
just- without needing to be mod anything
exists, so it's still an open question. And so
the research will continue. I mean the
greatest minds in mathematics will
continue to turn their attention, using
things like finite fields, and any
mathematical tool at their disposal
in the sole- like the greatest search in
maths is:
are numbers Parker,
or are numbers non-Parker?
This discussion continues for a bit
longer in a bonus video on our extras
channel Numberphile2.
That's- what is that- oh my goodness we
need a Parker prize. Well what are the
other prizes out there? There's like
Millennium, Fields - that's a
bit rich.
For more about what Matt Parker's up to,
from live shows to books to his own
YouTube channel, Stand-up Maths, check out
the various links in the video
description. - Could we even communicate
with beings from a non-Parker reality? It
boggles the mind.