Uncle Petros and Goldbach

by Apostolos Doxiadis

  Ironically, these would also be his last published work.

  After this first collaboration Hardy, an uncompromising judge of mathematical talent, proposed to Petros that he accept a fellowship at Trinity and settle in Cambridge to become a permanent part of their elite team.

  Petros asked for time to think it over. Of course, the proposal was enormously flattering and the prospect of continuing their collaboration had, at first glance, great appeal. Continued association with Hardy and Littlewood would no doubt result in more fine work, work that would assure his rapid ascent in the scientific community. In addition, Petros liked the two men. Being around them was not only agreeable but enormously stimulating. The very air they breathed was infused with brilliant, important mathematics.

  Yet, despite all this, the prospect of staying on filled him with apprehension.

  If he remained in Cambridge he would steer a predictable course. He would produce good, even exceptional work, but his progress would be determined by Hardy and Littlewood. Their problems would become his own and, what's worse, their fame would inevitably outshine his. If they did manage eventually to prove the Riemann Hypothesis (as Petros hoped they would) it would certainly be a feat of great import, a world-shaking achievement of momentous proportions. But would it be his? In fact, would even the third of the credit due to him by right be truly his own? Wasn't it likelier that his part in the achievement would be eclipsed by the fame of his two illustrious colleagues?

  Anybody who claims that scientists – even the purest of the pure, the most abstract, high-flying mathematicians – are motivated exclusively by the Pursuit of Truth for the Good of Mankind, either has no idea what he's talking about or is blatantly lying. Although the more spiritually inclined members of the scientific community may indeed be indifferent to material gains, there isn't a single one among them who isn't mainly driven by ambition and a strong competitive urge. (Of course, in the case of a great mathematical achievement the field of contestants is necessarily limited – in fact, the greater the achievement the more limited the field. The rivals for the trophy being the select few, the cream of the crop, competition becomes a veritable gigantomachia, a battle of giants.) A mathematician's declared intention, when embarking on an important research endeavour, may indeed be the discovery of Truth, yet the stuff of his daydreams is Glory.

  My uncle was no exception – this he admitted to me with full candour when recounting his tale. After Berlin and the disappointment with 'dearest Isolde' he had sought in mathematics a great, almost transcendent success, a total triumph that would bring him world fame and (he hoped) the cold-hearted Mädchen begging on her knees. And to be complete, this triumph should be exclusively his own, not parcelled out and divided into two or three.

  Also weighing against his staying on in Cambridge was the question of time. Mathematics, you see, is a young man's game. It is one of the few human endeavours (in this very similar to sports) where youth is a necessary requirement for greatness. Petros, like every young mathematician, knew the depressing statistics: hardly ever in the history of the field had a great discovery been made by a man over thirty-five or forty. Riemann had died at thirty-nine, Niels Henrik Abel at twenty-seven and Evariste Galois at a mere tragic twenty, yet their names were inscribed in gold in the pages of mathematical history, the 'Riemann Zeta Function', 'Abelian Integrals' and 'Galois Groups' an undying legacy for future mathematicians. Euler and Gauss may have worked and produced theorems into advanced old age, yet their fundamental discoveries had been made in their early youth. In any other field, at twenty-four Petros would be a promising beginner with years and years and years of rich creative opportunities ahead of him. In mathematics, however, he was already at the peak of his powers.

  He estimated that he had, with luck, at the most ten years in which to dazzle humanity (as well as 'dearest Isolde') with a great, magnificent, colossal achievement. After that time, sooner or later, his strength would begin to wane. Technique and knowledge would hopefully survive, yet the spark required to set off the majestic fireworks, the inventive brilliance and the sprightly spirit-of-attack necessary for a truly Great Discovery (the dream of proving Goldbach's Conjecture was by now increasingly occupying his thoughts) would fade, if not altogether disappear.

  After not-too-long deliberation he decided that Hardy and Littlewood would have to continue on their course alone.

  From now on he couldn't afford to waste a single day. His most produetive years were ahead of him, irresistibly urging him forward. He should immediately set to work on his problem.

  As to which problem this would be: the only candidates he had ever considered were the three great open questions that Caratheodory had casually mentioned a few years back – nothing smaller would suit his ambition. Of these, the Riemann Hypothesis was already in Hardy and Littlewood's hands and scientific savoir-faire, as well as prudence, deemed that he leave it alone. As to Fermat's Last Theorem, the methods traditionally employed in attacking it were too algebraic for his taste. So, the choice was really quite simple: the vehicle by which he would realize his dream of fame and immortality could not be other than Goldbach's humble-sounding Conjecture.

  The offer of the Chair of Analysis at Munich University had come a bit earlier, at just the right moment. It was an ideal position. The rank of full professor, an indirect reward for the military usefulness of the Papachristos Method to the Kaiser's army, would grant Petros freedom from an excessive teaching load and provide financial independence from his father, should he ever get the notion of attempting to lure him back to Greece and the family business. In Munich, he would be practically free of all irrelevant obligations. His few lecture hours would not be too much of an intrusion on his private time; on the contrary they could provide a constant, living link with the analytic techniques he would be using in his research.

  The last thing Petros wanted was to have others intruding on his problem. Leaving Cambridge, he had deliberately covered his tracks with a smokescreen. Not only did he not disclose to Hardy and Littlewood the fact that he would henceforth be working on Goldbach's Conjecture, but he led them to believe that he would be continuing work on their beloved Riemann Hypothesis. And in this too, Munich was ideal: its School of Mathematics was not a particularly famous one, like that of Berlin or the near-legendary Göttingen, and thus it was safely removed from the great centres of mathematical gossip and inquisitiveness

  In the summer of 1919, Petros settled in a dark second-floor apartment (he believed that too much light is incompatible with absolute concentration) at a short walk from the university. He got to know his new colleagues at the School of Mathematics and made arrangements regarding the teaching programme with his assistants, most of them his seniors. Then he set up his working environment in his home, where distractions could be kept to a minimum. His housekeeper, a quiet middle-aged Jewish lady widowed in the recent war, was told in the most unambiguous manner that once he had entered his study he was not to be disturbed, for any reason on earth.

  After more than forty years, my uncle still remembered with exceptional clarity the day when he began his research.

  The sun had not yet risen when he sat at his desk, picked up his thick fountain pen and wrote on a clean, crisp piece of white paper:

  STATEMENT: Every even number greater than 2 is the sum oftwo primes.

  PROOF: Assume the above Statement to be false. Then, there is an integer n such that 2n cannot be expressed as the sum oftwo primes, i.e.for every prime p ‹ In, 2n-p is composite…

  After a few months of hard work, he began to get a sense of the true dimensions of the problem and sign-posted the most obvious dead-ends. He could now map out a main strategy for his approach and identify some of the intermediate results that he needed to prove. Following the military analogy, he referred to these as the 'hills of strategie importance that had to be taken before mounting the final attack on the Conjecture itself'.

  Of course, his whole approach was based on the analytic me
thod.

  In both its algebraic and its analytic versions, Number Theory has the same object, namely to study the properties of the integers, the positive whole numbers 1,2,

  3,4,5… etc as well as their interrelations. As physical research is often the study of the elementary particles of matter, so are many of the central problems of higher arithmetic reduced to those of the primes (integers that have no divisors other than 1 and themselves, like 2, 3,5, 7,11…), the irreducible quanta of the number system.

  The Ancient Greeks, and after them the great mathematicians of the European Enlightenment such as Pierre de Fermat, Leonard Euler and Carl Friedrich Gauss, had discovered a host of interesting theorems concerning the primes (of these we mentioned earlier Euclid's proof of their infinitude). Yet, until the middle of the nineteenth century, the most fundamental truths about them remained beyond the reach of mathematicians.

  Chief among these were two: their 'distribution' (i.e. the quantity of primes less than a given integer n), and the pattern of their succession, the elusive formula by which, given a certain prime p_{n}, one could determine the next, p_{n+1}. Often (maybe infinitely often, according to a hypothesis) primes come separated by only two integers, in pairs such as 5 and 7, 11 and 13, 41 and 43, or 9857 and 9859. [6] Yet, in other instances, two consecutive primes can be separated by hundreds or thousands or millions of non-prime integers – in fact, it is extremely simple to prove that for any given integer k, one can find a succession of k integers that doesn't contain a single prime [7].

  The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedevilled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?

  The analytic theory of numbers was born in 1837, with Dirichlet's striking proof of the infinitude of primes in arithmetic progressions. Yet it didn't reach its peak until the end of the century. Some years before Dirichlet, Carl Friedrich Gauss had arrived at a good guess of an 'asymptotic' formula (i.e. an approximation, getting better and better as n grows) of the number of primes less than a certain integer n. Yet neither he nor anyone after him had been able to suggest a hint of a proof. Then in 1859, Bernhard Riemann introduced an infinite sum in the plane of complex numbers, [8] ever since known as the 'Riemann Zeta Function', which promised to be an extremely useful new tool. To use it effectively, however, number theorists had to abandon their traditional, algebraic (so-called 'elementary') techniques and resort to the methods of Complex Analysis, i.e. the infinitesimal calculus applied to the plane of complex numbers.

  A few decades later, when Hadamard and de la Vallee-Poussin managed to prove Gauss's asymptotic formula using the Riemann Zeta Function (a result henceforth known as the Prime Number Theorem) the analytic approach suddenly seemed to become the magic key to the innermost secrets of Number Theory.

  It was at the time of this high tide of hope in the analytic approach that Petros began his work on Goldbach's Conjecture.

  After spending the initial few months familiarizing himself with the dimensions of his problem, he decided he would proceed through the Theory of Partitions (the different ways of writing an integer as a sum), another application of the analytic method. Apart from the central theorem in the field, by Hardy and Ramanujan, there also existed a hypothesis by the latter (another of his famous 'hunches') which Petros hoped would become a crucial stepping stone to the Conjecture itself – if only he managed to prove it.

  He wrote to Littlewood, asking as discreetly as possible whether there had been any more recent developments in this matter, his question purportedly expressing 'a colleague's interest'. Littlewood reported in the negative, also sending him Hardy's new book, Some Famous Problems of Number Theory. In it, there was a proof of sorts of what is known as the Second or 'other' Conjecture of Goldbach [9]. This so-called proof, however, had a fundamental lacuna: its validity relied on the (unproven) Riemann Hypothesis. Petros read this and smiled a superior smile. Hardy was becoming pretty desperate, publishing results based on unproven premises! Goldbach's main Conjecture, the Conjecture, as far as he was concerned, was not even given lip service; his problem was safe.

  Petros conducted his research in total secrecy, and the deeper his probing led him into the terra incognita defined by the Conjecture, the more zealously he covered his tracks. For his more curious colleagues he had the same decoy answer that he'd used with Hardy and Littlewood: he was building on the work he had done with them in Cambridge, continuing their joint research on the Riemann Hypothesis. With time, he became cautious to the point of paranoia. In order to avoid his colleagues' drawing conclusions from the items he withdrew from the library, he began to find ways of disguising his requests. He would protect the book he really wanted by including it in a list of three or four irrelevant ones, or he would ask for an article in a scientific journal only in order to get his hands on the issue that also contained another article, the one he really wanted, to be perused far from inquiring eyes in the total privacy of his study.

  In the spring of that year, Petros received an additional short communication from Hardy, announcing Srinivasa Ramanujan's death of tuberculosis, at the age of thirty-two, in a slum neighbourhood of Madras. His first reaction to the sad news perplexed and even distressed him. Under a surface layer of sorrow for the loss of the extraordinary mathematician and the gentle, humble, sweet-spoken friend, Petros feit deep inside a wild joy that this phenomenal brain was no longer in the arena of Number Theory.

  You see, he had feared no one else. His two most qualified rivals, Hardy and Littlewood, were too involved with the Riemann Hypothesis to think seriously about Goldbach's Conjecture. As to David Hubert, generally acknowledged to be the world's greatest living mathematician, or Jacques Hadamard, the only other number theorist to be reckoned with, both were by now really no more than esteemed veter-ans – their almost sixty years were tantamount to advanced old age for creative mathematicians. But he had feared Ramanujan. His unique intellect was the only force he considered capable of purloining his prize. Despite the doubts he had expressed to Petros about the general validity of the Conjecture, should Ramanujan ever have decided to focus his genius on the problem… Who knows, maybe he would have been able to prove it despite himself; maybe his dear goddess Namakiri would have offered the solution to him in a dream, all neatly written out in Sanskrit on a roll of parchment!

  Now, with his death, there was no longer any real danger of someone arriving at the solution before Petros.

  Still, when he was invited by the great School of Mathematics at Göttingen to deliver a memorial lecture on Ramanujan's contribution to Number Theory, he carefully avoided mentioning his work on Partitions, lest anyone be inspired to look into its possible connections with Goldbach's Conjecture.

  In the late summer of 1922 (as it happened, on the very same day that his country was ravaged by the news of the destruction of Smyrna) Petras was suddenly faced with his first great dilemma.

  The occasion was a particularly happy one: while taking a long walk on the shore of the Speichersee, he arrived by way of a sudden illumination, following months of excruciating work, at an amazing insight. He sat down in a small beer-garden and scribbled it in the notebook he always carried with him. Then he took the first train back to Munich and spent the hours of dusk till dawn at his desk, working out the details and going over his syllogism carefully, again and again. When he was finished he felt for the second time in his life (the first had to do with Isolde) a feeling of total fulfilment, absolute happiness. He had managed to prove Ramanujan's hypothesis!

  In the first years of his work on the Conjecture, he had accumulated quite a few interesting intermediate results, so-called 'lemmas'
or smaller theorems, some of which were of unquestionable interest, ample material for several worthwhile publications. Yet he had never been seriously tempted to make these public. Although they were respectable enough, none of them could qualify as an important discovery, even by the esoteric standards of the number theorist.

  But now things were different.

  The problem he had solved on his afternoon walk by the Speichersee was of particular importance. As regarded his work on the Conjecture it was of course still an intermediate step, not his ultimate goal. Nevertheless, it was a deep, pioneering theorem in its own right, one which opened new vistas in the Theory of Numbers. It shed a new light on the question of Partitions, applying the previous Hardy-Ramanujan theorem in a way that no one had suspected, let alone demonstrated, before. Undoubtedly, its publication would secure him recognition in the mathematical world much greater than that achieved by his method for solving differential equations. In fact, it would probably catapult him to the first ranks of the small but select international Community of number theorists, practically on the same level as its great stars, Hadamard, Hardy and Littlewood.

  By making his discovery public, he would also be opening the way into the problem to other mathematicians who would build on it by discovering new results and expand the limits of the field in a way a lone researcher, however brilliant, could scarcely hope. The results they would achieve would, in turn, aid him in his pursuit of the proof to the Conjecture. In other words, by publishing the ‘Papachristos Partition Theorem' (modesty of course obliged him to wait for his colleagues formally to give it this title) he would be acquiring a legion of assistants in his work. Unfortunately there was another side to this coin: one of the new unpaid (also unasked for) assistants might conceivably stumble upon a better way to apply his theorem and manage, God forbid, to prove Goldbach's Conjecture before him.