Can I tell you about a new number?
It's a very scary number, it's a bit evil. It's Belphegor's prime.
Okay, so Belphegor, is one of the seven princes of hell. And the reason there's seven princes is because there are, of course, seven deadly sins.
He's also Hell's ambassador to France.
I didn't know Hell had ambassadors to different countries,
makes you wonder who Hell's ambassador to England is, right, but-
[mumbles] it's Nigel Farage, isn't it?
Anyway, so what is Belphegor's prime? Okay, so let's - he has a number.
[Ominous music]
So this number here is Belphegor's prime. So you can already see the
evil lurking within there, right? Number of the beast: 666 sitting square in the middle.
But it's even more evil than this. Thirteen zeros on either side of the 666, of course thirteen being a classic
unlucky number. And the other thing that's evil about this is that there's actually thirty-one digits;
and of course thirty one is thirteen backwards, isn't it? So this is a, this is a
fundamentally evil number. It's one of a larger class of numbers which are called the Belphegor numbers.
So a Belphegor number is basically anything - so B of n,
so the nth Belphegor number is a 1,
with n zeros,
666,
and then another n zeroes, and then a 1, okay?
So any Belphegor number has this form. If you want to be fancy about it and write a nice formula for it,
the formula you would write is this: 10 to the n plus 3 plus 666,
times 10 to the n plus 1,
plus 1.
These are the Belphegor numbers. Now, the vast majority of them are not prime, okay. So which ones are prime?
Well, of course for n equals 13, it's a prime number, because that's our classic Belphegor
prime. n equals zero is also a prime number. So that's of course this guy,
right, and this is also another example of a Belphegor Prime, this is the classic one
but this is another one. And this is quite interesting as well
for other reasons.
So this is the
1928th prime number, okay?
So this is a prime index. And if you take the numbers from here and add them together,
so 1 plus 6 plus 6 plus 6 plus 1,
well that's 20.
And you take these numbers and you add them together, 1 plus 9 plus 2 plus 8, well, that's also 20. So that's weird, okay?
(Brady: That's, that's tenuous.)
Do you reckon? Ok, well, let's not stop - we're gonna get more tenuous Brady, don't worry about that.
So this is also, this one here as well,
this particular Belphegor prime, is also a Republican prime. A Republican prime is any number
which is essentially prime to the right.
Okay? So it could be a Republican prime, in England I suppose it'd be a Tory prime.
Okay, so it's prime to the right. So you take the right-hand side of the number, not counting the middle,
so in this case here, 61, and that's a prime number. So it's kind of a coincidence
that - well I don't know what it says about Donald Trump that,
this Belphegor prime is also a Republican prime but we'll leave you to speculate on that.
(Brady: Do you get Democratic primes, or like, Labour primes?) - You do get democratic primes,
yes, you do. That's of course if you take it to the left, yes.
Of this is not; this is clearly not a democratic prime because 16 is not a prime number.
This set of numbers, these Belphegor numbers, and the Belphegor primes, were discovered by the famous prime number hunter Harvey Dubner.
Who is basically - he's an amateur number theorist. He built a massive computer in his garage with his son and
he just hunts for prime numbers.
And this was one of the ansatzes that he shoved into his computer to see if he could find examples of
prime numbers. He came up with this one. There are a few others,
there are more obviously. n equals 42 is one, which is pretty interesting because that's of course the
meaning of life, right? n equals 506 is the next one - and there's a whole tower of these, okay. Are there an infinite number?
Well, we don't know, and the reason we definitely don't know is because this is an example of a palindromic prime, so it obviously
looks the same forward and backward, and we don't know if there are an infinite number of palindromic primes so we certainly don't know if there's an
infinite number of Belphegor primes. It's also a naughty prime.
Obviously, not just because it's Belphegor. It's a naughty prime,
a naughty prime is one that has more zeros than any other number,
and of course this example does, so it's also a naughty prime.
(Brady: I've never heard of that! That's brilliant!)
Naughty primes? Yeah, they're pretty cool, aren't they? Well, it's lots of things. It's a beastly prime!
Okay, so a beastly prime is one that has a 666 in it.
Now this one has it in a very special place, right in the middle, right?
But a beastly prime is one that has 666.
So, for example another beastly prime is this one, 6661, 26669
is a beastly prime - there's a whole bunch of these as well. There's loads of these beastly primes, anything with, with 666 in it.
(Brady: We're not finished yet,)
(if you want to know even more about this evilness,)
(Check out the extra video on Numberphile2.)
[Preview] This is, I think, quite scary as well,
and this one hasn't got a name as far as I know.
(Brady: And there'll be a bit of a twist.)
I didn't want this to be all about evil right?
So I thought we should try to redress the balance with a bit of good.
(Brady: But if you'd like to know more about)
(666 itself - of course, we've done a video about that in the past. Check out links on the screen and down in the video description.)