31 and Mersenne Primes - Numberphile


Uploaded by numberphile on 09.01.2012

Transcript:

JAMES GRIME: OK, so today we're going to
talk Mersenne numbers.
So we have to pick one of our Mersenne numbers to exemplify
the category.
Now, Brady always makes me do this, so I've decided to pick
31, because by the time you watch this
video, I will be 31.
So it's a nice personal number for me.
So 31 is our example of a Mersenne prime.

So a Mersenne prime is a prime number to begin with like 31.
31 is a prime number.
A prime number is, as you probably know, a number that
can only be divided by 1 and itself.
The first few prime numbers are 2, 3, 5, 7.
31 is a prime number, but it's a special type of prime number
because it's one less than a power of 2.
What are the powers of 2?
Well, you've got 2.
You've got 4.
You've got 8, 16, 32, and so on.
OK, so that's just 2 times 2 times 2 times 2 times 2.
That's the sort of thing.
None of these are prime numbers, but let's have a look
at one less.
Well, 1.
Now, if I'm honest, we don't say 1 is prime.
There's a special reason for that.
But let's have a look at the others.
3 is a prime number, and it's one less than a power of 2, so
that counts as a Mersenne prime.
7 is a prime number, and it's one less than a power of 2.
Fantastic.
15.
Now that's not a prime number, so that does not count.
And 31, there you go, that was our example
of a Mersenne prime.
Now, there aren't many of these Mersenne primes.
In fact, there's only 47 Mersenne primes.
Now, they start off as 3, 7, 31.
The next one after that is 127.
So let's have a look at the next couple.
Powers of 2--
64, 128, and, well, this is 63, and that's not prime.
But this is 127.
That is prime, so it's one of our Mersenne primes.
They're called Mersenne primes because they're named after a
French mathematician called Marin Mersenne, who was a
monk, a mathematician, and a musician, and a fan of
alliteration.
His favorite cartoon was Mighty Mouse.
His favorite film was Mad Max.
So he was in contact with a lot of other mathematicians
around the world.
And he was trying to make a list of this special type of
prime number, and that's why they're named after him.
MALE SPEAKER: What are they used for?
Are they used by code breakers?
Are they used to make better iPads?
Or are they just a game for smart mathematicians like you?
JAMES GRIME: Well, prime numbers are obviously very
useful to mathematicians.
They're famous, and they're famous for a reason.
Mersenne primes are a special types of prime number.
Something very special about Mersenne primes, they are
related to the perfect numbers.
We've talked about perfect numbers before.
They were 6, 28, 496, 8,128.
They were known by the Ancient Greeks.
They were given this idea that they were perfect, unique.
Mersenne primes and the perfect numbers are two sides
of the same coin.
Let me show you why.
If I pick a Mersenne prime, let's call it M
for Mersenne prime.
OK, so M is our Mersenne prime.
Times it by M plus 1, divide by 2, and that will give you a
perfect number.
Let me show you.
3 times 4 divide by 2.
That's 12 divided by 2.
That's 6.
That was the first perfect number.
OK, let's try the next one--
7.
7 times 8, divide by 2.
That's 56 divide by 2.
That's 28.
That's the second perfect number.
31?
Same thing.
31 times 32, divide by 2, and that will give you the third
perfect number--
496.
And let's try 127.
127 times 128.
And half it, it's 8,128.
Two sides of the same coin.
Right, so you can help to find the next Mersenne prime.
It's called The Great Internet Mersenne Prime Search.
A great website with an unfortunate acronym.
It's at mersenne.org, and you can try it yourself.
Download the program.
It uses collaborative computer power to find the next
Mersenne prime.
MALE SPEAKER: Why?
Why should I do that?
Why should anyone do that?
JAMES GRIME: For the glory.
For the glory.
Because mathematicians will thank you for it.
Like the perfect numbers, we don't know if there's
infinitely many perfect numbers.
We don't know if there are infinitely many Mersenne
primes or not.
So that's another open question, something that
maybe, hey, you could work out.
MALE SPEAKER: What's it going to take to work that out?
How will that question ever be answered?
JAMES GRIME: It's not going to be easy.
It's not going to be obvious.
It's going to take a different way of looking at things,
coming at it from a different angle, something that other
people haven't tried yet.
Maybe not your usual school math, but something else.
Whatever it is, something we haven't tried yet.
MALE SPEAKER: The man or woman who does that, what will it
mean for them?
If a man or a woman comes up with that
proof, what will happen?
JAMES GRIME: If you do you come up with that proof, you
will have glory thrust upon you.
You will be famous for coming up with such a proof,
something that has eluded mathematicians
for hundreds of years.
MALE SPEAKER: Is it something you can have a hunch about?
JAMES GRIME: Yeah, I think technically we would have to
be Mersenne prime agnostic, but it's definitely something
you have a hunch for.
Mathematics is done that way.
People think mathematics is a dry subject, that is logical.
But people have intuition.
They have to pursue their intuition.
So if you ask me, I would say yeah, infinite number of
Mersenne primes?
Yeah, I would think so.
I might be wrong.