We've been so lucky on Numberphile
of having Ron Graham talking about Graham's number.
It was amazing that we could do that.
And Graham's number was a famous number,
that was the largest number, at that time, it was the largest number that have ever been used
in a proper mathematical proof.
And then we had got Ron Graham to talk about it. It's fantastic
"12 is not enough, it turns out,
and this other, slightly bigger number."
But before Ron Graham, the number, the largest number that had been used in a proper mathematical proof
was a different number called "Skews" number, or "Skewes" number?
It's a word that I've only ever seen written down, so I'm not quiet sure.
But it's a large number, and I'll show you what that number is.
10 to power 10, to power 10, to power 34.
Now this is going to be a large number.
This is a number that is something like trillions of trillions of trillions digits long.
I mean massive, I mean googol, a googol is a hundred digits long,
This is trillions and trillions and trillions of digits long.
Right, so this is huuuge.
So, where did this number come from?
So this number,
was like the Graham's number of its time...
Such a huge number!
I mean, even Issac Asimov wrote about this number.
So Isaac Asimov,
I've taken this from Brady, this is not mine, this is Brady's book,
I took it off his shelf.
Isaac Asimov, the, the, science fiction author,
wrote about this number as well, wrote an article about it,
in that kind of Numberphile-y kinda way.
If there was a Numberphile in the 1970s,
we would have had Isaac Asimov on, talking about Skewes number,
which would have been amazing.
Cameraman: That would be cool.
I know, 1970s Numberphile would've been awesome.
We would've had massive ties and sideburns,
and we'd be constantly smoking.
I wanna see that!
Compared to Graham's number, this is tiny.
But at that time, that this could-
I mean, I'm talking, this is before...
we've had computers and things,
this is a massively large number.
What was the problem they were solving with this number?
It was to do with primes and counting primes.
So the classic problem which goes way back, to
how many primes are there,
let's say, less than a given number, right?
So this goes back to Gauss.
I've talked about this before, the prime number theorem, which is fantastic stuff.
Primes less than x, which we write as pi of x.
So this is not pi. This is pi as in p for prime, rather than p for perimeter
which I think is what the, maybe what pi comes from.
So, if we do pi of 10, how many primes are there less than 10?
Two and three and five and seven. Tell me if I'm wrong, all right?
So there's four, so that would be four.
And if we did, how many primes less than 100?
I know this. There's 25 less than 100
And if I did how many primes are there less than, I don't know, a billion, then, less than a billiion?
That number is 50 847 534 primes less than that.
Now, Gauss, this, incredible, when he was fifteen years old,
and you think he'd have better things to do, hanging 'round the shops or something,
but when he was fifteen years old
he came up with a formula to approximate what this value might be.
His formula, originally, was, this number is round about x over the log of x.
That's the natural log of x. You might write that as ln of x.
Some of you may have seen that. So this would have been like this.
So this is an approximation, now, so, you know, it's a bit askew, it's not quite right.
They did come up with a better approximation. I want to show you the better approximation.
It was Dirichlet who came up with this, and his approximation
that the prime counting function is
something called, we'll call it Li of x,
it's called the logarithmic integral.
It's a function. Some of you will not be familiar with what I'm going to do,
which is fine, it's just a function, just say, you know, a clever maths formula for this.
If you are familiar with what I'm doing, then you'll understand what this is.
This is defined as the integral between two and x of one over the natural log of t dt.
But yes, a formula, so there's a formula for it.
So let's apply it, let's see what it does, let's do the same numbers, let's do 10.
So let's change that to a 10, okay?
If I do it for 10, I think the number you get is 5.
Which is not bad, okay? So it's not right, it's not correct, but it's, hey, it's not bad, 5, and it's 4.
If you do it for 100, you get 29.
It was actually 25, so you're 4 off. It's pretty good actually. It's pretty accurate.
And if we do it for the big number, and it gets more accurate the bigger the numbers are,
so let's do it for a billion, if we do it for a billion, how close are we going to get to the true value?
Which was 50 847 534, you get
50 849 234
You're so close, considering how big these numbers are,
that is 1700 off, which is not much when you're talking about the size of numbers.
So this is really quite accurate, it's really good.
But it does appear, and you can see there, that this estimation always seems to be bigger.
So always over-estimating. It's always bigger.
And Gauss had a look at this as well, he said, Gauss thought, it always seems to be bigger.
And he checked it, too. He checked it up to, like, three million, and it was always bigger.
So, people thought your prime counting function was always going to be less than
this estimation thing we're doing.
This is what they thought. Okay, that would be great.
Until.
Until
Littlewood, that's another mathematician. A Cambridge mathematician at the beginning of the 20th century.
We're talking about a hundred years ago, 1914
Littlewood realized that it's not always true.
That your logarithmic integral approximation sometimes isn't bigger. That and the quantity sign flips.
It flips, and in fact it does that infinitely many times
throughout the numbers, but he couldn't tell you the first time it flipped.
It must be something big, but he couldn't tell you the first time it did it.
So this mathematician Skewes found a approximation or a bound
for when the first time this inequality finally flips.
This number is his bound.
He said, he's seeing, it must be less than this.
It flips, and it flips at a value less than that number.
He doesn't know what it is yet, but he knows it's less than that.
Cameraman: "It could be a lot less."
It could be a lot less, exactly. It could be a lot less than that.
But he knows it's not bigger than that. So the first time it flips.
Now, we've improved this estimate since then. We now think that it might flip somewhere around
1.398 times ten to the power 316.
And actually it made me think this is actually quite a good estimate for Skewes Number,
the first time this flips.
So, we're talking a number that is a googol googol googol,
big number,
we don't know exactly where that flip is, we still haven't found where that flip is.
We're pretty sure it's somewhere around that area, though. We're pretty confident of that.
We have found a run of integers where the inequality is the other way round.
There is a run of integers between these numbers
1.53 times ten to the 1165
and 1.65 times ten to the power of 1165.
There is a run of consecutive integers where this inequality is the other way round, by which I mean,
your prime counting function is greater than the estimate.
This is a example of thinking that because a pattern holds for a very long time,
it's going to hold forever.
And it turns out, no.
So this is a pattern that holds, which we think for numbers going bigger and bigger and bigger
until they get into huge sizes of googols of googols of googols,
and it appears that this inequality holds and then it flips.
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