Читать книгу The Music of the Primes: Why an unsolved problem in mathematics matters - Marcus Sautoy du - Страница 11

The first to start telling these stories were the ancient Greeks. They realised the power of proof to forge permanent pathways to mountains in the mathematical world. Once they were reached, no longer was there the fear that these mountains were some distant mathematical mirage. For example, how can we be really sure that there aren’t some rogue numbers out there which can’t actually be built by multiplying together prime numbers? The Greeks were the first to come up with an argument that would leave no doubt in their minds or in the minds of future generations that no such rogue numbers could ever turn up.

Mathematicians often discover proofs by taking a particular instance of the general theory they are trying to prove, and begin by trying to understand why the theory is true for this example. They hope that the argument or recipe that was successful when applied to the example will work regardless of the particular case they chose to analyse. For instance, to prove that every number is a product of primes, start by considering the particular case of the number 140. Suppose you had checked that every number below 140 is either a prime number or the product of prime numbers multiplied together. What about the number 140 itself? Is it possible that this is a rogue number which is neither prime nor equal to a product of prime numbers? First, you would discover that the number is not prime. How would you do this? By showing it could be written as two smaller numbers multiplied together. For example, 140 is 4 × 35. Now we are ‘in’ because we have already confirmed that 4 and 35, numbers lower than our first candidate rogue, 140, can be written as primes multiplied together: 4 is 2 × 2 and 35 is 5 × 7. Piecing this information together, we see that 140 is in fact the product 2 × 2 × 5 × 7. So 140 is not a rogue after all.

The Greeks understood how they could translate this particular example into a general argument that would apply to all numbers. Curiously, their argument begins by asking us to imagine that there are such rogue numbers – ones that are neither prime nor can be written as prime numbers multiplied together. If there are such rogues, then, as we count through the sequence of all the numbers, we must eventually encounter the first of these rogue numbers. We shall call it N (it is sometimes referred to as the minimal criminal). Since this hypothetical number N isn’t a prime number, we must be able to write it as two smaller numbers, A and B, multiplied together. After all, if that weren’t possible, N would be prime.

Since A and B are smaller than N, our choice of N implies that A and B can be written as products of primes. So if we multiply together all the primes coming from A and all the primes coming from B, then we must get the original number, N. We have now shown that N can be written as prime numbers multiplied together, which contradicts our original choice of N. So our original assumption that there were rogue numbers can’t be tenable. Hence every number must be prime or built by multiplying primes.

When I tried this argument out on friends, they felt as if they had been cheated somewhere along the way. There is something slightly slippery about our opening gambit: assume the things you don’t want to exist do exist, and end up proving they don’t. This strategy of thinking the unthinkable became a powerful tool in the Greeks’ construction of proofs. It relies on the logical fact that a statement has to be either true or false. If we assume the statement is false and we get a contradiction, we can infer that our assumption was wrong and deduce that the statement must have been true after all.

The Greek proof appeals to the lazy side of most mathematicians. Instead of being faced with the impossible task of doing an infinite number of explicit calculations to prove that all numbers can be built from primes, the abstract argument captures the essence of every such computation. It’s like knowing how to climb an infinite ladder without physically having to perform the task.

More than any other Greek mathematician, Euclid is regarded as the father of the art of proof. He was part of the research institute that the Greek leader Ptolemy I established in Alexandria around 300 BC. There, Euclid wrote one of the most influential textbooks in all of recorded history: The Elements. In the first part of this book he set down axioms for geometry describing the relationship between points and lines. These axioms were put forward as self-evident truths about the objects of geometry, so that geometry would then act as a mathematical description of the physical world. He went on to use the rules of deduction to produce five hundred theorems of geometry.

The middle part of Euclid’s Elements deals with the properties of numbers, and it is here that we find what many regard as the first truly brilliant piece of mathematical reasoning. In Proposition 20, Euclid explains a simple but fundamental truth about prime numbers: there are infinitely many of them. He begins with the fact that every number can be built by multiplying prime numbers together. On top of this he constructs his next proof. If these prime numbers are the building blocks of all numbers, is it possible, he asks himself, for there to be only a finite number of these building blocks? The Periodic Table of the chemical elements was constructed by Mendeleev, and in its present form it classifies 109 different atoms from which we can build all matter. Could the same be true for prime numbers? What if a mathematical Mendeleev presented Euclid with a list of 109 primes and challenged Euclid to prove that some primes were missing from the list?

Why, for example, can’t all numbers be built simply by multiplying together different combinations of the primes 2, 3, 5 and 7? Euclid thought about how you might look for numbers that aren’t built from any of these primes. You might say, ‘Well that’s easy – just take the next prime, 11.’ This certainly can’t be built from 2, 3, 5 and 7. But sooner or later this strategy is going to fail precisely because, even today, we have no clue about how to guarantee finding where the next prime will be. And because of this unpredictability, Euclid had to try a different tack in his search for a method that would work, regardless of how long the list of primes was.

Whether it was truly Euclid’s own idea or whether he was simply recording ideas that others had dreamt up in Alexandria, we have no way of knowing. Whichever it was, he was able to show how to build a number that couldn’t be built from any finite list of primes that he might be given. Take the primes 2, 3, 5 and 7. Euclid multiplied them together to get 2 × 3 × 5 × 7 = 210, then – and this is his act of genius – he added 1 to this product to get 211. Euclid had constructed this number, 211, in such a way that none of the primes in the list, 2, 3, 5 and 7, would divide into it exactly. By adding 1 to the product, he could guarantee that dividing by a prime on the list would always leave remainder 1.

Now, Euclid knew that all numbers were built by multiplying together primes. So what about 211? Since it can’t be divided by 2, 3, 5 or 7, there had to be some other primes not on the list that built the number 211. In this particular example, 211 is itself a prime. Euclid was not claiming that the number he built would always be prime – only that it was a number that was built out of primes that were not on the list that our mathematical Mendeleev was offering us.

For example, what if one claimed that all numbers could be built from the finite list of primes 2, 3, 5, 7, 11 and 13. Euclid’s number built from these primes is 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031. This number is not a prime. All Euclid was saying was that, given any list containing finitely many primes, he could produce a number that had to be built out of primes that were not on the list. In this particular case the primes you need are 59 and 509. But in general, Euclid had no way of knowing how to find the precise value of these new primes. He knew only that they must exist.

It was a wonderful argument. Euclid had no idea how to produce primes explicitly, but he could prove why they would never run dry. It is striking that we do not know whether infinitely many of Euclid’s numbers themselves are prime, even though they are sufficient to prove that there must be an infinite number of primes. With Euclid’s proof, gone was the chance of fitting together a Periodic Table listing all the primes or of discovering some prime number genome coding billions of them. No simple butterfly collecting would ever allow us to understand these numbers. Here, then, was the ultimate challenge: the mathematician, armed with a limited weaponry, pitched against the infinite expanse of prime numbers. How could we possibly chart a path through such an infinite chaotic jumble and find some pattern which might predict their behaviour?