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 Goldbart’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, Goldbart’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.

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