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.
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