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

his son would join him in the church. The young Euler’s precocious mathematical talents, however, had brought him to the notice of the powers that be. Euler was soon being courted by the academies throughout Europe. He had been tempted to join the Academy in Paris, which by this time had become the world’s centre of mathematical activity. He chose instead to accept an offer made to him in 1726 to join the Academy of Sciences in St Petersburg, the capstone for Peter the Great’s campaign to improve education in Russia. He would be joining friends from Basel who had stimulated his interest in mathematics as a child. They wrote to Euler from St Petersburg asking whether he could bring from Switzerland fifteen pounds of coffee, one pound of the best green tea, six bottles of brandy, twelve dozen fine tobacco pipes and a few dozen packs of playing cards. Laden down with gifts, it took the young Euler seven weeks to complete the long journey by boat, on foot and by post wagon, and in May 1727 he finally arrived in St Petersburg to pursue his mathematical dreams. His subsequent output was so extensive that the St Petersburg Academy was still publishing material that had been housed in their archives some fifty years after Euler’s death in 1783.

      The role of the court mathematician is perfectly illustrated by a story that was told of Euler’s time in St Petersburg. Catherine the Great was hosting the famous French philosopher and atheist Denis Diderot. Diderot was always rather damning of mathematics, declaring that it added nothing to experience and served only to draw a veil between human beings and nature. Catherine, though, quickly tired of her guest, not because of Diderot’s disparaging views on mathematics but rather his tiresome attempts to rattle the religious faith of her courtiers. Euler was promptly called to court to assist in silencing the insufferable atheist. In appreciation of her patronage, Euler duly consented and addressed Diderot in serious tones before the assembled court: ‘Sir, (a + bⁿ)/n = x, hence God exists; reply.’ Diderot is reported to have retreated in the light of such a mathematical onslaught.

      This anecdote, told by the famous English mathematician Augustus De Morgan in 1872, had probably been embroidered for popular consumption and is a reflection more of the fact that most mathematicians enjoy putting down philosophers. But it does show how the royal courts of Europe had not considered themselves complete without a smattering of mathematicians amongst the ranks of astronomers, artists and composers.

      Catherine the Great was interested not so much in mathematical proofs of the existence of God, but rather in Euler’s work on hydraulics, ship construction and ballistics. The Swiss mathematician’s interests ranged far and wide over the mathematics of the day. As well as military mathematics, Euler also wrote on the theory of music, but ironically his treatise was regarded as too mathematical for musicians and too musical for mathematicians.

      One of his popular triumphs was the solution of the Problem of the Bridges of Königsberg. The River Pregel, known now as the Pregolya, runs through Königsberg, which in Euler’s day was in Prussia (it’s now in Russia, and called Kaliningrad). The river divides, creating two islands in the centre of the town, and the Königsbergers had built seven bridges to span it (see overleaf).

      It had become a challenge amongst the citizens to see if anyone could walk around the town, crossing each bridge once and only once, and return to their starting point. It was Euler who eventually proved in 1735 that the task was impossible. His proof is often cited as the beginning of topology, where the actual physical dimensions of the problem are not relevant. It was the network of connections between different parts of the town that was important to Euler’s solution, and not their actual locations or distances apart – the map of the London Underground illustrates this principle.

      It was numbers above all that captivated Euler’s heart. As Gauss would write:

      The peculiar beauties of these fields have attracted all those who have been active there; but none has expressed this so often as Euler, who, in almost every one of his many papers on number-theory, mentions again and again his delight in such investigations, and the welcome change he finds there from tasks more directly related to practical applications.

      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