a heady mix of Ph.D-level pure mathematics

and world debating championships,

or, as I like to say, "Hello, ladies. Oh yeah."

Didn't get much sexier than the Spence

at university, let me tell you.

It is such a thrill for a humble breakfast radio announcer

from Sydney, Australia, to be here on the TED stage

literally on the other side of the world.

And I wanted to let you know, a lot of the things you've heard

about Australians are true.

From the youngest of ages, we display

a prodigious sporting talent.

On the field of battle, we are brave and noble warriors.

What you've heard is true.

Australians, we don't mind a bit of a drink,

sometimes to excess, leading to embarrassing social situations. (Laughter)

This is my father's work Christmas party, December 1973.

I'm almost five years old. Fair to say,

I'm enjoying the day a lot more than Santa was.

But I stand before you today

not as a breakfast radio host,

not as a comedian, but as someone who was, is,

and always will be a mathematician.

And anyone who's been bitten by the numbers bug

knows that it bites early and it bites deep.

I cast my mind back when I was in second grade

at a beautiful little government-run school

called Boronia Park in the suburbs of Sydney,

and as we came up towards lunchtime, our teacher,

Ms. Russell, said to the class,

"Hey, year two. What do you want to do after lunch?

I've got no plans."

It was an exercise in democratic schooling,

and I am all for democratic schooling, but we were only seven.

So some of the suggestions we made as to what

we might want to do after lunch were a little bit impractical,

and after a while, someone made a particularly silly suggestion

and Ms. Russell patted them down with that gentle aphorism,

"That wouldn't work.

That'd be like trying to put a square peg through a round hole."

Now I wasn't trying to be smart.

I wasn't trying to be funny.

I just politely raised my hand,

and when Ms. Russell acknowledged me, I said,

in front of my year two classmates, and I quote,

"But Miss,

surely if the diagonal of the square

is less than the diameter of the circle,

well, the square peg will pass quite easily through the round hole."

(Laughter)

"It'd be like putting a piece of toast through a basketball hoop, wouldn't it?"

And there was that same awkward silence

from most of my classmates,

until sitting next to me, one of my friends,

one of the cool kids in class, Steven, leaned across

and punched me really hard in the head.

(Laughter)

Now what Steven was saying was, "Look, Adam,

you are at a critical juncture in your life here, my friend.

You can keep sitting here with us.

Any more of that sort of talk, you've got to go and sit

over there with them."

I thought about it for a nanosecond.

I took one look at the road map of life,

and I ran off down the street marked "Geek"

as fast as my chubby, asthmatic little legs would carry me.

I fell in love with mathematics from the earliest of ages.

I explained it to all my friends. Maths is beautiful.

It's natural. It's everywhere.

Numbers are the musical notes

with which the symphony of the universe is written.

The great Descartes said something quite similar.

The universe "is written in the mathematical language."

And today, I want to show you one of those musical notes,

a number so beautiful, so massive,

I think it will blow your mind.

Today we're going to talk about prime numbers.

Most of you I'm sure remember that six is not prime

because it's 2 x 3.

Seven is prime because it's 1 x 7,

but we can't break it down into any smaller chunks,

or as we call them, factors.

Now a few things you might like to know about prime numbers.

One is not prime.

The proof of that is a great party trick

that admittedly only works at certain parties.

(Laughter)

Another thing about primes, there is no final biggest prime number.

They keep going on forever.

We know there are an infinite number of primes

due to the brilliant mathematician Euclid.

Over thousands of years ago, he proved that for us.

But the third thing about prime numbers,

mathematicians have always wondered,

well at any given moment in time,

what is the biggest prime that we know about?

Today we're going to hunt for that massive prime.

Don't freak out.

All you need to know, of all the mathematics

you've ever learned, unlearned, crammed, forgotten,

never understood in the first place,

all you need to know is this:

When I say 2 ^ 5,

I'm talking about five little number twos next to each other

all multiplied together,

2 x 2 x 2 x 2 x 2.

So 2 ^ 5 is 2 x 2 = 4,

8, 16, 32.

If you've got that, you're with me for the entire journey. Okay?

So 2 ^ 5,

those five little twos multiplied together.

(2 ^ 5) - 1 = 31.

31 is a prime number, and that five in the power

is also a prime number.

And the vast bulk of massive primes we've ever found

are of that form:

two to a prime number, take away one.

I won't go into great detail as to why,

because most of your eyes will bleed out of your head if I do,

but suffice to say, a number of that form

is fairly easy to test for primacy.

A random odd number is a lot harder to test.

But as soon as we go hunting for massive primes,

we realize it's not enough

just to put in any prime number in the power.

(2 ^ 11) - 1 = 2,047,

and you don't need me to tell you that's 23 x 89.

(Laughter)

But (2 ^ 13) - 1, (2 ^ 17) - 1

(2 ^ 19) - 1, are all prime numbers.

After that point, they thin out a lot.

And one of the things about the search for massive primes

that I love so much is some of the great mathematical minds

of all time have gone on this search.

This is the great Swiss mathematician Leonhard Euler.

In the 1700s, other mathematicians said

he is simply the master of us all.

He was so respected, they put him on European currency

back when that was a compliment.

(Laughter)

Euler discovered at the time the world's biggest prime:

(2 ^ 31) - 1.

It's over two billion.

He proved it was prime with nothing more

than a quill, ink, paper and his mind.

You think that's big.

We know that (2 ^ 127) - 1

is a prime number.

It's an absolute brute.

Look at it here: 39 digits long,

proven to be prime in 1876

by a mathematician called Lucas.

Word up, L-Dog.

(Laughter)

But one of the great things about the search for massive primes,

it's not just finding the primes.

Sometimes proving another number not to be prime is just as exciting.

Lucas again, in 1876, showed us (2 ^ 67) - 1,

21 digits long, was not prime.

But he didn't know what the factors were.

We knew it was like six, but we didn't know

what are the 2 x 3 that multiply together

to give us that massive number.

We didn't know for almost 40 years

until Frank Nelson Cole came along.

And at a gathering of prestigious American mathematicians,

he walked to the board, took up a piece of chalk,

and started writing out the powers of two:

two, four, eight, 16 --

come on, join in with me, you know how it goes --

32, 64, 128, 256,

512, 1,024, 2,048.

I'm in geek heaven. We'll stop it there for a second.

Frank Nelson Cole did not stop there.

He went on and on