TONY: I've got another constant for you Brady.
It's Apéry's constant.
One point two oh two oh five six nine ...
We could go on.
What is this really? This is actually - what I'm writing down here -
is really 1 + 1 over 2 cubed + 1 over 3 cubed ...
This sum actually appears in quantum electrodynamics.
It's related to the anomalous magnetic moment of the electron, which is one of the great tests of QED.
This number appears in that so it has physical relevance,
but that's not really why we're going to talk about it today.
The reason I want to talk about it is because it's -
It's a crazy number in many ways
and not a lot was known about it for a very long time, and it's still not known about.
This goes back to 1644 when Pietro Mengoli posed the Basel problem.
Of course, the Basel problem was - the question was not quite this sum
but essentially this sum
1 plus 1 over 2 squared, 1 over 3 squared and so on...
To get this in closed form - this was the problem, to get it in closed form
and this was famously solved by Euler in 1735
and he showed that this was equal to π squared over 6.
Now of course, Euler being Euler,
you know, wasn't going to settle for just this guy, right?
He wanted to do this far more generally
so he looked at the general form of these things: 1 over 2 to the s, 1 over 3 to the s
1 over 4 to the s and so on which we recognize as the
Riemann Zeta function, you know, one can ask what values this takes in the positive integers?
And this is what Euler was interested in and he was able to show of course that when you take even integers -
so you look at this guy at even values - that this took the form of something rational.
times π to the 2n
So of course, π being transcendental, this guarantees that this number is also going to be transcendental.
That was for the even positions.
What about the odds? The things like zeta to the 2n + 1 and which of course
This guy which is of course
our Apéry's constant is at the value three. What about this? Is this rational? Is this transcendental?
What, what is it? Euler couldn't work it out, Euler couldn't do it and nobody knew for hundreds of years
What was going on with this number.
Until this mathematician came along, a kind of maverick mathematician -
a good mathematician, but not a great mathematician, a guy called Roger Apéry.
actually solved this problem.
And Apéry's quite an interesting character actually:
He's French, he was very political, very politically active
He was a prisoner of war in the 2nd World War
and he sort of had quite a dry sense of humor actually.
Later on he was a member of the French resistance.
At one point he was actually stopped by the Gestapo because he was carrying an object -
A long thin object wrapped in newspaper.
And the Gestapo said to him: what are you carrying there? Is it a gun?
And he said no, it's a leg.
And it was a leg, it was his friend's prosthetic leg, that he was going to get fixed
So he was there, but you know, this was the Gestapo he was messing with, right?
So he was pushing his luck somewhat.
Anyway this sort of sense of humor - this sort of maverick style - actually carried through.
Let's fast-forward to 1978 - he's coming towards the end of his career
there's this - he's a good mathematician, but not a great one,
and he gives a seminar about the value of this zeta function at three, okay?
And whether it's rational or not which hadn't been and which the great Euler himself had not been able to solve.
Anyway, Apéry says he solved it. Okay. It is irrational, and he can prove it
So he starts his - and everyone's a bit skeptical, okay?
So he starts his seminar and the first thing he writes down is the following.
Doesn't really matter, details of that don't matter
The point is nobody in the audience has ever seen this formula before, okay?
It's not being proven, nothing.
So they immediately ask, you know, where did - no one's seen this before - where did they find it?
Where did he find this formula? And he says - he replies - "Oh, they grow in my garden."
So everybody's getting annoyed at this point. People start to leave and like, you know, everyone's furious,
and then while this is all going on - people are leaving -
some guy at the back's got a programmable calculator,
and manages to verify this sum, at least to the accuracy of the calculator,
and then, you know, as the talk's going on, he interrupts and says and announces this,
at which point everyone sits up and starts paying attention.
And it turns out that Apéry has solved this hundreds of years old problem,
and he proves that it is indeed irrational.
And of course it is fabulous - it sort of shook maths at the time.
People started to try to use it to then start to ask questions about the other odd values - you know -
zeta 5, zeta 7, and so on.
And they try to use the same method, and all methods like that have so far failed,
interestingly enough, and so it's still not known whether those guys are irrational or not.
BRADY: But three we cracked. TONY: Three is definitely irrational,
but it's not known whether it's transcendental.
What is known about the other odd ones is that there are an infinite number of irrational ones,
but we don't know which ones they are.
It's even known that one of zeta 5, zeta 7, zeta 9, and zeta 11,
that one of these is irrational, but we don't know which one.
BRADY: (laughs) TONY: So it's so little known about it and
It's sort of - I think it's a fabulous number.
So coming back to Apéry - I did a shout-out on Twitter.
I asked people to send in three randomly chosen positive integers,
and the reason I did that was 'cause I wanted to estimate Apéry's constant.
Now I'm a bit lazy, so I didn't go through all of them 'cause I've got loads!
And some people put ones that were a little bit hard to manipulate as well.
There's quite some quite entertaining comments about how you can't choose that random numbers from just 140 characters
And so on and so forth, but anyway
I did go through someone I went through quite a few.
I have a list here, and what I did was I went through those numbers,
and I checked whether the combinations were co-prime. So in other words I asked
What was their greatest common divisor - of the three numbers -
and if that greatest common divisor was 1, then they're co-prime.
If it was some number higher than 1 then they're not co-prime.
So I just went through and I checked them all, all the different combinations.
So let me do this - I've gotta count this now, Brady!
I haven't actually checked this - I didn't check the number - [if] the estimate came out right,
so this is real-time, real-action stuff.
So anyway the number that are co-prime -
Let's just - let's just count them, okay
So here's my list, so that's one - all the ticks are co-prime, okay.
BRADY: These are all the tweeters?!
TONY: These are all the tweeters - not all of them 'cause there were just too many, so I had to stop at some point,
but it's a lot of them, okay?
So these are all the tweeters - so anyway, okay,
So if they're ticked, then they're co-prime - they sent in co-prime combinations. So one, two,
three, four, five, six, seven, eight, nine,
ten, eleven, twelve, thirteen, fourteen, fifteen, ...
38, 39, 40, 41, 42,
43, 44, 45, ... 68, 69, 70, 71, 72.
So there's 72 of those came in as co-primes, co-prime combinations.
Some people didn't send in co-prime combinations
What I've written down here
are the greatest common divisors that they had so we'll just count how many of those there were.
One, two, three, four, five, ... fourteen, fifteen.
All right, did I miss any?
Okay, so that's a total of - it was 87 in total, right?
Okay, now I claim, that -
This is the moment of truth now, I dunno if this is going to work -
That 87 divided by the number of - so the total number divided by the co-prime number - let's see what number this gives.
(excited laughing)
It's not bad, not bad!
Is that good?
BRADY: It's pretty close. TONY: I think it's pretty close. I'm happy with that.
I'm very happy with that estimate. Okay, good.
Yay, and I think, I'm sure if I'd gone through all of them, this would've got more and more accurate, right?
Okay, how? Why?
Let's suppose we had s numbers.
What's the probability that any given number - so we got some prime number p -
What's the probability that any given - you pick a random number - that it's divisible by p?
It's 1 over p. If I've got s numbers, what's the probability that they're all divisible by p?
Well that's going to be 1 over p to the s.
What's the probability that at least one of those s numbers is not divisible by p?
That's 1 minus 1 over p to the s.
So if they - if I now ask,
whether if I've got s numbers,
whether the probability that their lowest [*greatest] common divisor is 1,
the basis - there's no number of the prime numbers that divides them all,
then I just have to take the product of this number over all the primes.
So that's the product over the primes
of 1 minus 1 over p to the s and now our friend Euler comes back, right?
Because it's all Euler - he's always [like?] that, Euler - Euler showed that the zeta function
Was the product over the primes
1 minus 1 over p to the s to the minus 1.
So this is 1 over zeta of s.
Okay so this means you've got a 1 in zeta s chance of finding a co-prime combination.
Okay, so for three -
I've got three numbers. The chances of finding a co-prime combination are 1 in zeta 3.
That's why it worked. Makes sense?
(outro music)
Thanks for watching this video. If you'd like to show a bit more support to Numberphile
You can support us on Patreon - the address is on the screen and in the video description.
That support helps us make even more and better videos
And we try to make it worth our patrons while by adding a few extra bits and pieces for them on our Patreon page.
But as always the best support you can give us is to be subscribed to Numberphile and keep watching.
We'll see you in the next video!