The Music of the Primes: Why an unsolved problem in mathematics matters. Marcus Sautoy du

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      The bridges of Konigsberg.

      Euler’s passion for number theory had been stimulated by correspondence with Christian Goldbach, an amateur German mathematician who was living in Moscow and unofficially employed as secretary of the Academy of Sciences in St Petersburg. Like the amateur mathematician Mersenne before him, Goldbach was fascinated by playing around with numbers and doing numerical experiments. It was to Euler that Goldbach communicated his conjecture that every even number could be written as a sum of two primes. Euler in return would write to Goldbach to try out many of the proofs he had constructed to confirm Fermat’s mysterious catalogue of discoveries. In contrast to Fermat’s reticence in keeping his supposed proofs a secret from the world, Euler was happy to show off to Goldbach his proof of Fermat’s claim that certain primes can be written as the sum of two squares. Euler even managed to prove an instance of Fermat’s Last Theorem.

      Despite his passion for proof, Euler was still very much an experimental mathematician at heart. Many of his arguments flew close to the mathematical wind, containing steps that weren’t completely rigorous. That did not concern him if it led to interesting new discoveries. He was a mathematician of exceptional computational skill and very adept at manipulating mathematical formulas until strange connections emerged. As the French academician François Arago observed, ‘Euler calculated without apparent effort, as men breathe, or eagles sustain themselves in the wind.’

      Above all else, Euler loved calculating prime numbers. He produced tables of all the primes up to 100,000 and a few beyond. In 1732, he was also the first to show that Fermat’s formula for primes, , broke down when N = 5. Using new theoretical ideas, he managed to show how to crack this ten-digit number into a product of two smaller numbers. One of his most curious discoveries was a formula that seemed to generate an uncanny number of primes. In 1772, he calculated all the answers that you get when you feed the numbers from 0 to 39 into the formula x² + x + 41. He got the following list:

      41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1,033, 1,097, 1,163, 1,231, 1,301, 1,373, 1,447, 1,523, 1,601

      It seemed bizarre to Euler that you could generate so many primes with this formula. He realised that the process would have to break down at some point. It might already be clear to you that when you input 41, the output has to be divisible by 41. Also, for x = 40 you get a number which is not prime.

      Nonetheless, Euler was quite struck by his formula’s ability to produce so many primes. He began to wonder what other numbers might work instead of 41. He discovered that in addition to 41 you could also choose q = 2, 3, 5, 11, 17, and the formula x² + x + q would spit out primes when fed numbers from 0 to q − 2.

      But finding such a simple formula for generating all the primes was beyond even the great Euler. As he wrote in 1751, ‘There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.’ It seems paradoxical that the fundamental objects on which we build our order-filled world of mathematics should behave so wildly and unpredictably.

      It would turn out that Euler had been sitting on an equation that would break the prime number deadlock. But it would take another hundred years, and another great mind, to show what Euler could not. That mind belonged to Bernhard Riemann. It was Gauss, though, who by initiating another of his classic lateral moves, would eventually inspire Riemann’s new perspective.

      Gauss’s guess

      If centuries of searching had failed to unearth some magic formula which would generate the list of prime numbers, perhaps it was time to adopt a different strategy. This was what the fifteen-year-old Gauss was thinking in 1792. He had been given a book of logarithms as a present the previous year. Until a few decades ago, tables of logarithms were familiar to every teenager doing calculations in the schoolroom. With the advent of pocket calculators, they lost their place as an essential tool in everyday life, but several hundred years ago every navigator, banker and merchant would have been exploiting these tables to turn difficult multiplication into simple addition. Included at the back of Gauss’s new book was a table of prime numbers. It was uncanny that primes and logarithms should appear together, because Gauss noticed after extensive calculations that there seemed to be a connection between these two seemingly unrelated topics.

      The first table of logarithms was conceived in 1614 in an age when sorcery and science were bedfellows. Their creator, the Scottish Baron John Napier, was regarded by local residents as a magician who dealt in the dark arts. He skulked around his castle dressed in black, a jet-black cock perched on his shoulder, muttering that his apocalyptic algebra foretold that the Last Judgement would fall between 1688 and 1700. But as well as applying his mathematical skills to the practice of the occult, he also uncovered the magic of the logarithm function.

      If you feed a number, say 100, into your calculator and then press the ‘log’ button, the calculator spits out a second number, the logarithm of 100. What your calculator has done is to solve a little puzzle: it has looked for the number x that makes the equation 10^x = 100 correct. In this case the calculator outputs the answer 2. If we input 1,000, a number ten times larger than 100, then the new answer output by your calculator is 3. The logarithm goes up by 1. Here is the essential character of the logarithm: it turns multiplication into addition. Each time we multiply the input by 10, we get the new output by adding 1 to the previous answer.

      It was a fairly major step for mathematicians to realise that they could talk about logarithms of numbers which weren’t whole-number powers of 10. For example, Gauss would have been able to look up in his tables the logarithm of 128 and find that raising 10 to the power 2.107 21 would get him pretty close to 128. These calculations are what Napier had collected together in the tables that he had produced in 1614.

      Tables of logarithms helped to accelerate the world of commerce and navigation that was blossoming in the seventeenth century. Because of the dialogue that logarithms created between multiplication and addition, the tables helped to convert a complicated problem of multiplying together two large numbers into the simpler task of adding their logarithms. To multiply together large numbers, merchants would add together the logarithms of the numbers and then use the log tables in reverse to find the result of the original multiplication. The increase in speed that the sailor or seller would gain via these tables might save the wrecking of a ship or the collapse of a deal.

      But it was the supplementary table of prime numbers at the back of his book of logarithms that fascinated the young Gauss. In contrast to the logarithms, these tables of primes were nothing more than a curiosity to those interested in the practical application of mathematics. (Tables of primes constructed in 1776 by Antonio Felkel were considered so useless that they ended up being used for cartridges in Austria’s war with Turkey!) The logarithms were very predictable; the primes were completely random. There seemed no way to predict when to expect the first prime after 1,000, for example.

      The important step Gauss took was to ask a different question. Rather than attempting to predict the precise location СКАЧАТЬ