If you look at the current top ten in
the prime charts you'll see the top 10
largest primes at the moment, and you may
notice that one of them stands out. One
of them is not quite like the others.
It's quite a special prime, it was
actually discovered fairly recently; it was
discovered on October 31st 2016, it
was discovered on Halloween 2016. But all
the other primes are Mersenne primes, they're all
primes that are one less than a power of
two. And there's a reason for that,
because computationally it's quite easy
to check if they are primes or not, right?
There is a- there is a test for that
which you can run quite quickly. Number
seven there is slightly different.
There's a couple of reasons why it's
special and one of the reasons is that
it takes us one step closer to solving a
fifty-year-old conjecture. So let's take
a look at this prime. So this number- it's
called a Proth prime. So Proth prime's
have this form where it's k, that's
some number, multiplied by a power of two
plus one. Let's see if we can find a
Proth prime. What I'm gonna do, let's take a
value; I'm gonna say k is 19, all
right? Let's take k as 19 and then we're
gonna multiply it by powers of two, add
one, and let's see if we get a prime. Well
let's do the first power of two. 19 times
2, 38. Plus 1 that equals 39, but that's not
a prime; it's 3 times 13 that's no good.
Let's check for another. Let's try 19
again but let's do 2 squared. So 19 times
times 4, well 19 times 2 is 38 so we can
double that again. That's 76, plus 1 that's 77;
that's definitely not a prime, that's 7
times 11. Let's keep going, let's
see if we can find one. Let's do 19 times 2
to power 3. That is 153,153 is that prime?
That's maybe not so obvious, but no
that isn't prime.
19 times 2 to the power 4, ok we're gonna
keep looking. I I think that is - plus 1 of
course - that is 305, that's definitely not
a prime. Okay another failure, let's keep
looking for a prime. Let's do the
next one. 19 times 2 to the 5 plus 1,
I've got that as being 609, no that's
divisible by 3. - (Brady: I think you're gonna get us)
(on that top 10 list at this rate! - It's gonna-
actually we've got it. We've got it - look at this one.
19 times 2 to the power 6,1217 - prime! That
is a prime. Finally we found a pri- took a
while. Yeah, it took a while but
eventually we found a prime. In the 1960s,
mathematician Sierpiński said that if
you take something of this form: it's k
times 2 to power n plus 1, and you keep
running through it, there are numbers that
will never hit a prime. So you'll keep
going forever and you'll never get
a prime. In fact he said there were
infinitely many of them. So there's loads of
them out there. The natural question is,
what's the smallest one? Ah, well we don't
quite know. So they think it might be
this number: 78557. - (Brady: As k?)
Yeah that's as k. So I'm saying if we multiply
that by 2 to the n, and then we add 1;
we run through that process we'll never
hit a prime, it's never going to happen.
In fact I know that's never going to
happen. So a mathematician called John
Selfridge showed that that's never gonna
happen. He showed that this number,
whatever it is,
will always be divisible by one of
these: 3, 5, 7, 13, 19, 37 or 73. So he said
it's always gonna be divisible by one of
those, so it's never going to be a prime.
That definitely is a Sierpiński number,
okay? This is a number that will never
hit a prime, great; but is it the smallest
one? That we don't know. Now there's quite
a few candidates for this - well in fact
we're gonna have to check all the
numbers smaller than 78557. Well you
can do that. And most of those were
eliminated. Until roundabout the
beginning of the 21st century there were
17 left. 17 that hadn't been eliminated.
So there might be numbers that are
smaller than 78557, and you could do
this and you'll never hit a prime. So
there were 17 candidates left. But, thanks
to advances
in computation, that meant we could
start eliminating these candidates. Now
we started to eliminate them until there
were six left. These are the six that
were left. The smallest was
10223. But here
are the others: 21181,
22699,
24737,
55459 and 67607.
So there were these six candidates left. But
things started to slow down and I'm
afraid there was a gap of nine years
before we were able to eliminate one of
these six, which was this one here:
10223
because we now know that forms a prime.
Here's, here's our prime, this is it! There
it is, we know it does form a prime. And
now this is struck off the list leaving
five left. Now if we can eliminate those
last five candidates we will have proved
that 78558 is the smallest
Sierpiński number. - (Brady: So James, was this)
(really large prime not so much found)
(because they were searching for large)
(primes but because they were trying to)
(eliminate ks?) - They- that is part of the
motivation for this. This was found
by a guy in Hungary on his home computer.
Where he's downloaded a program, you can
get it from the PrimeGrid website. And
this program was designed to eliminate
those seventeen that if I talked about. And
using that program he finally eliminated
10223.
So you can do this yourself, using
your own home computer. So he gets the
glory of solving that and the glory of
striking off one of these candidates off
our list. So we have these candidates and
then you find a prime that kills it off
as a candidate - so those primes are huge.
This one that was found recently is over
nine million digits
long, it's a massive number. The
largest of these primes that are killing
off candidates, they've actually got a
special name. They're called Colbert primes,
named after Stephen Colbert. Stephen
Colbert, some people might know, as the
host of The Late Show in America?
American comedian? I don't know why these
have been named after Stephen Colbert; I
guess someone's a fan. But if we can
eliminate all these candidates we will
have proven that 78557 is the smallest
Sierpiński number. We don't know it's
true, it just feels true; in our guts
it has truthiness.
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