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Talking primes

Updated Thursday 12th July 2012

What is so magical about prime numbers - and what makes them musical? The Material World wanted to find out...

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Quentin Cooper Copyrighted image Icon Copyright: BBC

Quentin Cooper:
Hello and welcome to prime time radio; we’ll be counting the ways in which prime numbers have fascinated, frustrated and been fundamental to mathematics for millennia. Why? Well, there’s any number of reasons, just as long as they are divisible by 1 and themselves.

Harken to a porky prime cut and think about what quality it suggests.

A piece of music is played

Quentin Cooper:
Something ‘elegiac’, sombreness, maybe a soft waft of cosmic futility, small child let loose in orchestra pit. If you thought ‘timelessness’ then you can feel fabulously in tune with the composer, Olivier Messiaen, because that was exactly what he was after. And when he was ‘messiaening’ about writing the piece, he hit upon using prime numbers as a way to evoke it.

There is something about primes - 2, 3, 5, 7, 11, 13 and every other number divisible only by 1 and themselves - that defies time and intuition and the efforts of the finest minds on the planet. As you’d expect for something given the name ‘prime’, they’re one of the foundations of mathematics. But although it’s well over 2,000 years since Euclid proved there’s an infinite number of them there’s still no formula for predicting primes, despite the strong incentive of a million dollar reward for whoever comes up with one. And as more than prize money as motivation, prime numbers - once an archetypally abstract subject of study - now hold the key to internet security, password encryption, and even natural phenomena [such as] the cycle by which cicadas appear by the millions as they did this year, spreading chaos across the Eastern United States.

To discuss why primes are now a prime concern within and beyond mathematics, I’m joined by Robin Wilson, head of Pure Maths at the Open University and Gresham Professor of Geometry, and by Marcus du Sautoy, Professor of Mathematics at Oxford, presenter of BBC FOUR’s Mind Games and Arsenal fan. I’m sure there’s some interesting analysis that can be applied to why 49 games without defeat is followed by 3 defeats and 5 draws in 11 games, but we will save that for another day.

Marcus, this Messiaen music now, how exactly did he use primes? I couldn’t exactly spot them there from what I heard.

Marcus du Sautoy:
Well, Messiaen was obsessed with bird song and well as mathematics so you probably heard some bird song …

Quentin Cooper:
He was obsessed with all sorts of … he was obsessed with cosmic rhythm and all sorts of things, he was a very obsessive chap.

Marcus du Sautoy:
Yes, prime numbers was something very dear to his heart and he actually used the primes in order to create a sense of timelessness in this piece. There is a 17 note sequence, and a 29 note sequence that he plays one after the other, and they mesh in different ways because they are both prime numbers, so you hear the 17 and 29 start off together and the 17 starts again after the 17 notes and the 29 is still going on. You only hear them repeat the pattern again after 17 times 29 times that you’ve heard the whole sequence. So the primes are being used by Massiaen to create the sense of timeless[ness] in the piece.

Quentin Cooper:
I must stress that this wasn’t audible from the little snatch we played then. I’ve said what a prime is, can you do the trickier job of explaining why it is so fascinating to yourself and to other mathematicians?

Marcus du Sautoy:
Absolutely, the primes are really the building blocks of the whole of mathematics. If you take a number like 105, it’s not a prime number, but if you divide and divide it, you get to 3 times 5 times 7 which makes 105. So they are really like the atoms of arithmetic, they’re like the hydrogen and oxygen of the world of mathematics.

Quentin Cooper:
But hang on, couldn’t you say that you can take the numbers 1 to 10, everything can be divided down to something that’s a multiple of those.

Marcus du Sautoy:
Not at all. Because if I take 23 I can’t divide that …

Quentin Cooper:
Apart from primes... Ah, I see the flaw in my argument.

Robin Wilson, what’s the attraction and fascination of primes for you?

Robin Wilson:
Well as you said, any number can be broken down into its atoms, its primes, and so in some sense, primes, as Marcus said, are the building blocks of the whole of numbers, and if you are interested in the multiplication of primes then you can get all the numbers. But the fascination to me, and I’ve had this fascination since I was a little boy, is actually trying to bring in also addition and subtraction of primes.

There’s a very famous problem called Goldbach's Conjecture. If you take any even number, it looks as though you can write it as a sum of two primes – 6 is 3 plus 3, 8 is 3 plus 5, 10 is 5 plus 5 or 7 plus 3. And it looks as though this goes on all the way up.

Quentin Cooper:
That seems reasonable …

Robin Wilson:
But here you’re trying to bring in primes, which are essentially things you’re meant to multiply, and you’re bringing in addition and subtraction which are really nothing to do with primes; and if you do that, then you get some of the hardest problems in mathematics. So this problem, Goldbach's Conjecture, [asks] can every even number be written as a sum of two primes? No-one knows the answer. It’s been checked up to 400 trillion, and for a scientist that would be lots and lots of evidence, but to a mathematician, truth is absolute, if there’s a single one that goes wrong, then everything’s...

Quentin Cooper:
Marcus, this is one of the things that mathematics runs into several times, however logical, however reasonable, however almost common sense, some things might seem, however bleedin’ obvious, that’s actually not enough, you’ve got to come up with billions and trillions of examples. We want something that says reliably 'this is a formula' and it predicts it, and it will do it until the end of time.

Marcus du Sautoy:
Yes, exactly. The point is, in mathematics, a lot of evidence, a million pieces of data, actually when you’re talking about an infinite number of numbers, represents an infinitely small proportion of them. So evidence is often misleading in mathematics. Proof is everything.

Quentin Cooper:
And is this part of the appeal of prime numbers, that they seem so simple, but within them, they are so unpredictable?

Marcus du Sautoy:
Well, that’s it, I describe a mathematician as a pattern searcher, that’s what I do all my life, is to try and look for pattern and logic. Yet our subject seems to be built on a set of numbers which have no patterns to them at all. This is really the ultimate tease for the mathematician, you know, nature’s given us these numbers which have no patterns and the subject is dedicated to finding patterns.

 

Marcus du Sautoy goes for a punt Copyrighted image Icon Copyright: Production team Quentin Cooper:
And you’d be perfectly happy as mathematicians, both of you, wouldn’t you, if you could prove that there definitely was no pattern, it’s just the suspicion that there might be a pattern but you can’t prove that it’s there, or that there is definitely no pattern and you can’t prove that either?

 

Marcus du Sautoy:
Exactly. I mean, we’d be very …. In fact the greatest unsolved problem in mathematics is something called the Riemann Hypothesis, is precisely to show that primes seem randomly scattered throughout all the numbers. And that’s quite a hard thing to do, to prove that there is some sort of randomness in the primes, it would be much easier to show that there’s a pattern.

Robin Wilson:
In fact, if you look at primes, you write out a list of all the primes, and of course you can’t because it goes on for ever, as you said, Euclid proved that, but if you look close to, you can see there are pairs, primes differing by just 2 – 3 and 5, 41 and 43, 101 and 103 – the further up you go, you still find them. Again, this is one of those easy to state problems, no-one knows whether that’s always the case.

We do know if we stand a long way away from the primes, you can see that they seem to have a general pattern, they seem to be thinning out, but if you look locally, you see primes very close together, but you can also get arbitrary long gaps between primes.

So this is the fascination of the primes, that locally you can’t see any pattern, you stand away, then you can, and this is this Riemann problem. Riemann explained in a particular way why the primes seem to thin out, but in doing so, he came up with a particular problem, which is now called the Riemann Hypothesis, and this problem, the Riemann Hypothesis, is generally accepted as being the most famous unsolved problem in the whole of mathematics.

Quentin Cooper:
Marcus, he did this, what, about 150 years ago …?

Marcus du Sautoy:
1859, yes.

Quentin Cooper:
…. And somewhere along the line it’s acquired this million dollar price tag?

Marcus du Sautoy:
That’s right, yes, actually most mathematicians would forget the money, we’d sell our souls to Mephistopholes for a proof of this, but …

Quentin Cooper:
I love this idea of mathematicians as creatures not of this Earth, like intellectual skylarks twittering a bit, not interested in the money at all!

Marcus du Sautoy:
Not interested in the money at all, that’s it. I mean the interesting thing is, what Riemann did was... I mean, mathematicians are very good at laterally thinking, changing a problem, looking at it in a new way, and actually what he discovered was a pattern in a completely different area of mathematics which explains why the primes look so random. So there’s a very strange tension here; that’s what Riemann discovered, and we can’t prove that what he discovered is really true, and that’s what the million dollars is really there for: to prove that Riemann was right about his hunch. But it’s really the triumph of the mathematician over nature, the pattern searchers found a pattern somewhere else, which would explain why the primes look so random.

Quentin Cooper:
Robin, what’s your hunch about Reimann’s hunch, will it be proved?

Robin Wilson:
I think it will be proved. It’s been linked in with lots of other areas. I mean, if it’s proved, it’s going to be quite important in mathematics, because it has all sorts of implications. Also, knowledge about the primes is of interest in other areas as well, like cryptography, which is more Marcus’s area than mine, I think there is some feeling that … well, the problem was actually stated by David Hilbert, as one of his unsolved problems in 1900. He was asked to give a big talk at a big international congress and he set the agenda for mathematics in the 20th century by saying 'here are 23 problems I would like to see solved this century'. Most of them were. The one that still sticks out is the Riemann Hypothesis. And that’s why it became of one of seven problems with a million dollars on its head

Quentin Cooper:
Marcus, as Robin’s already mentioned, this is no longer just of interest to academicians or people who want a very hard way to get a million dollars, there are increasingly practical applications for prime numbers, particularly cryptography.

Marcus du Sautoy:
And the interesting thing is that we really don’t understand the primes well enough, that we can use our ignorance to do encryption. And so for example, to crack a code, on the internet, when you go and send your credit card securely across the internet, what you have to do is, given a number, try and find the prime atoms which built that number.

You see, the chemists have this wonderful thing called a spectrometer, you give it a molecule, it’ll tell you the atoms it's built out of. Well, mathematicians don’t have a magic prime number spectrometer which will tell you the primes which built the number, say, 126,619. Not a prime built out of two smaller primes, we don’t have a very fancy way to find those primes.

Quentin Cooper:
So in other words, if you’re one of the people using the code, if you’ve got one of the primes, it’s very easy to divide that into the big number and get the other prime, but it’s very hard for anybody else to untangle what the two primes are that make it up.

Marcus du Sautoy:
Exactly.

Quentin Cooper:
Now, just briefly, the cicadas. What on earth have they got to do with primes?

Marcus du Sautoy:
The cicadas are very interesting, they relate back to the Messiaen, in fact. There’s a very strange cicada in North America that has a curious life cycle, hides underground for 17 years, doing absolutely nothing, then after 17 years they emerge en masse into the forest, party away for six weeks, then all die.

Quentin Cooper:
It’s scenes of Messiaen.Always 17 years?

Marcus du Sautoy:
Seventeen. There’s another species of 13, another of 7.

Quentin Cooper:
Why?

Marcus du Sautoy:
So why the primes? They think there was a predator that used to appear in the forest and used to eat all the cicadas up. The cicada found that by choosing a prime number of years, it could keep out of synch, just like Messiaen’s themes, just as the 17 and 29 kept out of synch for so long. So by the cicada choosing a 17 year life cycle, it kept out of synch of the predator, who was appearing periodically in the forest, and managed to survive.

Quentin Cooper:
Robin, I don’t know if you can do this in ten seconds, but I said primes numbers 2, 3, 5, 7, why is 1 not a prime?

Robin Wilson:
Because if you break a number up into primes, you can only do it in one way, so 100 is 2 times [2 times] 5 times 5, but it’s not 2 times 2 times 5 times 5 times 1, we don’t want to have the extra ones in there as well.

Quentin Cooper:
I hope that’s completely clear. Right, Professors Robin Wilson, Marcus du Sautoy, thanks for that primes primer, having sublimated our prime allergies.

Editor's note: Due to an error in the transcription process, Goldbach's Conjecture was originally given as Goldbart's Conjecture.

 

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