Two

I went to Ekali on the second day after my arrival in Greece for the summer vacation. Not wanting to catch him unawares, I'd already arranged the meeting with Uncle Petros by correspondence. To continue with the judicial analogy, I'd granted him ample time to prepare his defence.

I arrived at the arranged time and we sat in the garden.

'So then, most favoured of nephews' (this was the first time he called me that), 'what news do you bring me from the New World?'

If he thought I'd let him pretend this was a mere social occasion, a visit by dutiful nephew to caring uncle, he was mistaken.

'So then, Uncle,' I said belligerently, 'in a year's time I'm getting my degree and I'm already preparing applications for graduate school. Your ploy has failed. Whether it is to your liking or not, I will be a mathematician.'

He shrugged his shoulders while raising the palms of his hands heavenwards in a gesture of inevitability.

"'He who is fated to drown will never die in his bed",' he intoned – a populär Greek proverb. 'Have you told your father? Is he pleased?'

'Why this sudden interest in my father?' I snarled. 'Was it he who put you up to our so-called "deal"? Was it his perverse idea to make me prove myself worthy by tackling Goldbach's Conjecture? Or do you feel so much in his debt for supporting you all these years that you repaid him by bringing his upstart son to heel?'

Uncle Petros accepted the blows under the belt without changing expression.

'I don't blame you for being angry,’ he said. 'Yet you have to try to understand. Although my method was indeed questionable, the motives were as pure as driven snow.'

I laughed scornfully. "There is nothing pure in having your failure determine my life!'

He sighed. 'You have time at your disposal?'

'As much as you want.'

'And you are seated comfortably?'

‘Perfectly.'

'Then listen to my story. Listen and judge for your-self.'


The Story of Petros Papachristos


I cannot pretend to remember as I write now the exact phrasing and expressions my uncle used on that summer afternoon, so many years ago. I have preferred to recreate his narrative in the third person, opting for completeness and coherence. Where memory failed me I consulted his extant correspondence with family and mathematical colleagues as well as the thick, leather-bound volumes of the personal diaries in which he traced the progress of his research.


Petros Papachristos was born in Athens in November 1895. He spent his early childhood in virtual isolation, the first-born of a self-made businessman whose sole concern was his work and a housewife whose sole concern was her husband.

Great loves are often born of loneliness, and this certainly seems to have been true of my uncle's lifelong affair with numbers. He discovered his particular aptitude for calculation early on and it didn't take long for it, for lack of other emotional diversions, to develop into a veritable passion. Even as a little boy, he filled his empty hours doing complicated sums, mostly in his head. By the time his two little brothers' arrival enlivened the household he was already so committed to his pursuit that no changes in family dynamics could distract him.

Petros' school, a religious institution run by French Jesuit brothers, upheld the Order's brilliant tradition in mathematics. Brother Nicolas, his first teacher, immediately recognized his bent and took him under his wing. With his guidance the boy began to cover material way beyond the capabilities of his classmates. Like most Jesuit mathematicians, Brother Nicolas specialized in (the already old-fashioned by that time) classical geometry. He spent his time contriving exercises which, although often ingenious and as a rule monstrously difficult, held no deeper mathematical interest. Petros solved them, as well as any others his teacher culled from the Jesuit maths books, with astonishing ease.

However, his particular passion from the very beginning lay in the Theory of Numbers, a field in which the brothers were not particularly knowledgeable. His undeniable talent together with constant practice since his earliest years had resulted in almost uncanny skills. When Petros, at the age of eleven, heard that every positive integer can be expressed as the sum of four squares, he astonished the good brothers by providing the breakdown of whatever number was suggested after only a few seconds of thought.

'What about 99, Pierre?' they'd ask.

'99 is equal to 8^2 plus 5^2 plus 3^2 plus 1^2,' he'd answer.

'And 290?'

'290 is equal to 12^2 plus 9^2 plus 7^2 plus 4^2' But how on earth can you do it so fast?' Petros described a method that seemed obvious to him, but to his teachers was difficult to understand and impossible to apply without paper, pencil and sufficient time. The procedure was based on leaps of logic that bypassed intermediate steps of calculation, clear evidence that the boy's mathematical intuition was already developed to an extraordinary degree.

After having taught him more or less everything they knew, when Petros was fifteen or so the brothers found themselves unable to answer their gifted pupil's constant flow of mathematical questions. It was then that the headmaster went to his father. Papachristos pere may not have had much time for his children, but he knew his duty where the Greek Orthodox faith was concerned. He had enrolled his eldest son in a school run by schismatic foreigners because it held prestige within the social elite to which he aspired to belong. When faced with the headmaster's proposal, however, that his son be sent to a monastery in France in order to further cultivate his mathematical talent, his mind immediately went to proselytism.

'The damn papists want to get their hands on my son,' he thought.

Still, despite lacking higher education, the elder Papachristos was anything but naive. Knowing from personal experience that one succeeds best in the field of endeavour one has a natural gift for, he had no desire to place any obstacles in his son's natural course. He asked around in the right circles and was informed of the existence, in Germany, of a great mathematician who also happened to belong to the Greek Orthodox persuasion, the renowned Professor Constantin Caratheodory. He immediately wrote to him for an appointment.

Father and son travelled together to Berlin, where Caratheodory received them in his office at the university, dressed like a banker. After a short chat with the father he asked to be left alone with the son. He led him to the blackboard, gave him a piece of chalk and questioned him. Petros solved integrals, calculated the sums of series and proved statements, as prompted. Then, once the esteemed professor had finished his examination, the boy reported his own discoveries: elaborate geometric constructions, complex algebraic identities and, particularly, observations regarding the properties of the integers. One of those was the following:

'Every even number greater than 2 can be written as a sum of two primes.'

'You surely can't prove that,’ said the famous mathematician.

'Not yet,’ answered Petros, 'although I'm sure it's a general principle. I've checked it up to 10,000!'

'What about the distribution of the prime numbers?' Caratheodory asked. 'Can you figure a way to calculate how many primes there are lesser than a given number n?'

'No,’ answered Petros, 'but as n approaches infinity the quantity gets very close to its ratio by the natural logarithm.'

Caratheodory gasped. 'You must have read that somewhere!'

'No, sir, it just seems a reasonable extrapolation from my tables. Besides, the only books at my school are about geometry.'

The Professor's previously stern expression now gave way to a beaming smile. He called Petros' father inside and told him that to subject his son to two more years of high school would be a complete waste of precious time. Denying his extraordinarily gifted boy the best that mathematical education had to offer would be tantamount, he said, to 'criminal negligence'. Caratheodory would arrange to have Petros immediately admitted to his university – if his guardian consented, of course.

My poor grandfather never had a choice: he had no desire to commit a crime, especially against his first-born.

Arrangements were made, and a few months later Petros returned to Berlin and moved into the family home of a business associate of his father 's, in Charlottenburg.

During the months preceding the start of the next academic year, the eldest daughter of the house, the eighteen-year-old Isolde, undertook to help the young foreign guest with his German. It being summer, the tutoring sessions were often conducted in secluded corners of the garden. When it got colder, Uncle Petros reminisced with a mellow smile, 'the instruction was continued in bed'.

Isolde was the first and (judging from his narrative) only love my uncle ever had. Their affair was brief and conducted in total secrecy. Their trysts would take place at irregular times in unlikely locations, at noon or midnight or dawn, in the shrubbery or the attic or the wine cellar, wherever and whenever the opportunity for invisibility beckoned: if her father found out, he would string him up by his thumbs, the girl had repeatedly warned her young lover.

For a while, Petros was totally disoriented by love. He became almost indifferent to everything other than his sweetheart, to the point that Caratheodory came to wonder for a while whether he might have been wrong in his original appreciation of the boy's potential. But after a few months of tortuous happiness ('alas, too few,' my uncle said with a sigh), Isolde abandoned the family home and the arms of her boy-lover in order to marry a dashing lieutenant of the Prussian artillery.

Petros was, of course, heartbroken.

If the intensity of his childhood passion for numbers was partly a recompense for the lack of familial tenderness, his immersion into higher mathematics at Berlin University was definitely made all the more complete for the loss of his beloved. The deeper he now delved into its endless ocean of abstract concepts and arcane symbols, the farther he was mercifully removed from the excruciatingly tender memories of 'dearest Isolde'. In fact, in her absence she became 'of much more use' (his words) to Petros. When they had first lain together on her bed (when she had first thrown him on to her bed, to be precise) she had softly muttered in his ear that what attracted her to him was his reputation as a Wunderkind, a little genius. In order to win her heart back, Petros now decided, there could be no half-measures. To impress her at a more mature age he should have to accomplish amazing intellectual feats, nothing short of becoming a Great Mathematician.

But how does one become a Great Mathematician? Simple: by solving a Great Mathematical Problem!

'Which is the most difficult problem in mathematics, Professor?' he asked Caratheodory at their next meeting, trying to feign mere academic curiosity.

‘I’ll give you the three main contenders,' the sage replied after a moment's hesitation. 'The Riemann Hypothesis, Fermat's Last Theorem and, last but not least, Goldbach's Conjecture, the proof of the observation about every even number being the sum of two primes – one of the great unsolved problems of Number Theory.'

Although by no means yet a firm decision, the first seed of the dream that some day he would prove the Conjecture was apparently planted in his heart by this short exchange. The fact that it stated an observation he had himself made long before he'd heard of Goldbach or Euler made the problem dearer to him. Its formulation had attracted him from the very first. The combination of external simplicity and notorious difficulty pointed of necessity to a profound truth.

At present, however, Caratheodory was not allowing Petros any time for daydreaming.

'Before you can fruitfully embark on original research,’ he told him in no uncertain terms, 'you have to acquire a mighty arsenal. You must master to perfection all the tools of the modern mathematician from Analysis, Complex Analysis, Topology and Algebra.'

Even for a young man of his extraordinary talent, this mastery needed time and single-minded attention.

Once he'd received his degree, Caratheodory assigned him for his doctoral dissertation a problem from the theory of differential equations. Petros surprised his master by completing the work in less than a year, and with spectacular success. The method for the solution of a particular variety of equations which he put forth in his thesis (henceforth, the 'Papachristos Method') earned him instant acclaim because of its usefulness in the solution of certain physical problems. Yet – and here I'm quoting him directly – 'it was of no particular mathematical interest, mere calculation of the grocery-bill variety.'

Petros was awarded his doctorate in 1916. Immediately afterwards, his father, worried about the imminent entry of Greece into the melee of the Great War, arranged for him to settle for a while in neutral Switzerland. In Zürich, at last a master of his fate, Petros turned to his first and constant love: numbers.

He sat in on an advanced course at the university, attended lectures and seminars, and spent all his remaining time at the library, devouring books and learned journals. Soon, it became apparent to him that to proceed as fast as possible to the frontiers of knowledge, he had to travel. The three mathematicians doing world-class work in Number Theory at that time were the Englishmen G. H. Hardy and J. E. Littlewood and the extraordinary self-taught Indian genius Srinivasa Ramanujan. All three were at Trinity College, Cambridge.

The war had divided Europe geographically, with England practically cut off from the mainland by patrolling German U-boats. However, Petros' intense desire, combined with his total indifference to the danger involved as well as his more than ample means, soon got him to his destination.

'I arrived in England still a beginner,’ he told me, 'but left it, three years later, an expert number theorist.'

Indeed, the time in Cambridge was his essential preparation for the long, hard years that followed. He had no official academic appointment, but his – or rather his father's – financial situation allowed him the luxury of subsisting without one. He settled down in a small boarding-house next to the Bishop Hostel, where Srinivasa Ramanujan was staying at the time. Soon, he was on friendly terms with him and together they attended G. H. Hardy's lectures.

Hardy embodied the prototype of the modern research mathematician. A true master of his craft, he approached Number Theory with brilliant clarity, using the most sophisticated mathematical methods to tackle its central problems, many of which were, like Goldbach's Conjecture, of deceptive external simplicity. At his lectures, Petros learned the techniques which would prove necessary to his work and began to develop the profound mathematical intuition required for advanced research. He was a fast learner, and soon he began to chart out the labyrinth into which he was fated soon to enter.

Yet, although Hardy was crucial to his mathematical development, it was his contact with Ramanujan that provided him with inspiration.

'Oh, he was a totally unique phenomenon,’ Petros told me with a sigh. 'As Hardy used to say, in terms of mathematical capability Ramanujan was at the absolute zenith; he was made of the same cloth as Archimedes, Newton and Gauss – it was even conceivable that he surpassed them. However, the near-total lack of formal mathematical training during his formative years had for all practical purposes condemned him never to be able to fulfil anything but a tiny fraction of his genius.'

To watch Ramanujan do mathematics was a humbling experience. Awe and amazement were the only possible reactions to his uncanny ability to conceive, in sudden flashes or epiphanies, the most inconceivably complex formulas and identities. (To the great frustration of the ultra-rationalist Hardy, he would often claim that his beloved Hindu goddess Namakiri had revealed these to him in a dream.) One was led to wonder: if the extreme poverty into which he had been born had not deprived Ramanujan of the education granted to the average well-fed Western student, what heights might he have attained?

One day, Petros timidly brought up with him the subject of Goldbach's Conjecture. He was purposely tentative, concerned that he might awaken his interest in the problem.

Ramanujan's answer came as an unpleasant surprise. 'I have a hunch, you know, that the Conjecture may not apply for some very very big numbers.'

Petros was thunderstruck. Could it possibly be? Coming from him, this comment couldn't be taken lightly. At the first opportunity, after a lecture, he approached Hardy and repeated it to him, trying at the same time to appear rather blase about the matter.

Hardy smiled a cunning little smile. 'Good old Ramanujan has been known to have some wonderful "hunches",' he said, 'and his intuitive powers are phenomenal. Still, unlike His Holiness the Pope, he lays no claim to infallibility.'

Then Hardy eyed Petros intently, a gleam of irony in his eyes. 'But tell me, my dear fellow, why this sudden interest in Goldbach's Conjecture?'

Petros mumbled a banality about his 'general interest in the problem' and then asked, as innocently as possible: 'Is there anyone working on it?'

'You mean actually trying to prove it?' said Hardy. 'Why no – to attempt to do so directly would be sheer folly!'

The warning did not scare him off; on the contrary it pointed out the course he should follow. The meaning of Hardy's words was clear: the straightforward, so-called 'elementary' approach to the problem was doomed to failure. The right way lay in the oblique 'analytic' method that, following the recent great success of the French mathematicians Hadamard and de la Vallee-Poussin with it, had become tres a la mode in Number Theory. Soon, he was totally immersed in its study.

There was a time, in Cambridge, before he made the final decision about his life's work, when Petros seriously considered devoting his energies to a different problem altogether. This came about as a result of his unexpected entry into the Hardy-Littlewood-Ramanujan inner circle.

During those wartime years, J. E. Littlewood had not been spending much time around the university. He would show up every now and then for a rare lecture or a meeting and then disappear once again to God knows where, an aura of mystery surrounding his activities. Petros had yet to meet him and so was greatly surprised when, one day in early 1917, Littlewood sought him out at the boarding-house.

'Are you Petros Papachristos from Berlin?' he asked him, after a handshake and a cautious smile. 'Constantin Caratheodory's Student?'

'I am the one, yes,’ answered Petros perplexed.

Littlewood appeared slightly ill at ease as he went on to explain: he was at that time in charge of a team of scientists doing ballistics research for the Royal Artillery as part of the war effort. Military intelligence had recently alerted them to the fact that the enemy's high accuracy of fire in the Western Front was thought to be the result of an innovative new technique of calculation, called the 'Papachristos Method'.

‘I’m sure you wouldn't have any objection to sharing your discovery with His Majesty's Government, old chap,' Littlewood concluded. 'After all, Greece is on our side.'

Petros was at first dismayed, fearing he would be obliged to waste valuable time with problems that held no more interest for him. That didn't prove necessary, though. The text of his dissertation, which he luckily had with him, contained more than enough mathematics for the needs of the Royal Artillery. Littlewood was doubly pleased since the Papachristos Method, apart from its immediate usefulness to the war effort, significantly lightened his own load, giving him more time to devote to his main mathematical interests.

So: rather than side-tracking him, Petros' earlier success with differential equations provided his entry into one of the most renowned partnerships in the history of mathematics. Littlewood was delighted to learn that the heart of his gifted Greek colleague belonged, as did his, to Number Theory, and soon he invited him to join him on a visit to Hardy's rooms. The three of them talked mathematics for hours on end. During this, and at all their subsequent meetings, both Littlewood and Petros avoided any mention of what had originally brought them together; Hardy was a fanatical pacifist and strongly opposed to the use of scientific discoveries in facilitating warfare.

After the Armistice, when Littlewood returned to Cambridge full-time, he asked Petros to collaborate with him and Hardy on a paper they had originally begun with Ramanujan. (The poor fellow was by now seriously ill and spending most of his time in a sanatorium.) At that time, the two great number theorists had turned their efforts to the Riemann Hypothesis, the epicentre of most of the unproven central results of the analytic approach. A demonstration of Bernhard Riemann's insight on the zeros of his 'zeta function' would create a positive domino effect, resulting in the proof of countless fundamental theorems of Number Theory. Petras accepted their proposal (which ambitious young mathematician wouldn't?) and the three of them jointly published, in 1918 and 1919, two papers – the two that my friend Sammy Epstein had found under his name in the bibliographical index.

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 method.

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.

He didn't have to deliberate long. The danger far outweighed the benefit. He wouldn't publish. The 'Papachristos Partitions Theorem' would remain for the time being his private, well-guarded secret.

Reminiscing for my benefit, Uncle Petros marked this decision as a turning point in his life. From then onwards, he said, difficulties began to pile upon difficulties.

By withholding publication of his first truly important contribution to mathematics, he had placed himself under double time-pressure. In addition to the constant, gnawing anxiety of days and weeks and months and years passing without his having achieved the desired final goal, he now also had to worry that someone might arrive at his discovery independently and steal his glory.

The official successes he had achieved until then (a discovery named after him and a university chair) were no mean feats. But time counts differently for mathematicians. He was now at the absolute peak of his powers, in a creative prime that couldn't last long.

This was the time to make his great discovery – if he had it in him to make it at all.

Living as he did a life of near-total isolation, there was no one to ease his pressures.

The loneliness of the researcher doing original mathematics is unlike any other. In a very real sense of the word, he lives in a universe that is totally inaccessible, both to the greater public and to his immediate environment. Even those closest to him cannot partake ot his joys and his sorrows in any significant way, since it is all but impossible for them to understand their content.

The only community to which the creative mathematician can truly belong is that of his peers; but from that Petros had wilfully cut himself off. During his first years at Munich he had submitted occasionally to the traditional academic hospitality towards newcomers. When he accepted an invitation, however, it was sheer agony to act with a semblance of normality, behave agreeably and make small talk. He had constantly to curb his tendency to lose himself in number-theoretical thoughts, and fight his frequent impulses to make a mad dash for home and his desk, in the grip of a hunch that required immediate attention. Fortunately, either as a result of his increasingly frequent refusals or his obvious discomfort and awkwardness on those occasions when he did attend social functions, invitations gradually grew fewer and fewer and in the end, to his great relief, ceased altogether.

I don't need to add that he never married. The rationale he gave me for this, by which getting married to another woman would mean being unfaithful to his great love, 'dearest Isolde', was of course no more than an excuse. In truth, he was very much aware that his lifestyle did not allow for the presence of another person. His preoccupation with his research was ceaseless. Goldbach's Conjecture demanded him whole: his body, his soul and all of his time.

In the summer of 1925, Petros proved a second important result, which in combination with the 'Partitions Theorem' opened up a new perspective on many of the classical problems of prime numbers. According to his own, exceedingly fair and well-informed opin-ion, the work he had done constituted a veritable breakthrough. The temptation to publish was now overwhelming. It tortured him for weeks – once again, though, he managed to resist it. Again, he decided in favour of keeping his secret to himself, lest it open the way to unwelcome intruders. No intermediate result, no matter how important, could sidetrack him from his original aim. He would prove Goldbach's Conjecture or be damned!

In November of that year he turned thirty, an emblematic age for the research mathematician, practically the first step into middle age.

The sword of Damocles, whose presence Petros had merely sensed all these years hanging in the darkness somewhere high above him (it was labelled: 'The Waning of his Creative Powers') now became almost visible. More and more, as he sat hunched over his papers, he could feel its hovering menace. The invisible hourglass measuring out his creative prime became a constant presence at the back of his mind, driving him into bouts of dread and anxiety. During his every waking moment, he was pestered by the worry that he might already be moving away from the apex of his intellectual prowess. Questions buzzed in his mind like mosquitoes: would he be having any more breakthroughs of the same order as the two first important results? Had the inevitable decline, perhaps unbeknown to him, already started? Every little instance of forgetfulness, every tiny slip in a calculation, every short lapse in concentration, brought the ominous refrain: Have I passed my prime?

A brief visit at about this time from his family (already described to me by my father), whom he hadn't seen in years, was considered by him a gross, violent intrusion. The little time he spent with his parents and younger brothers he felt was stolen from his work, and every moment away from his desk for their benefit he perceived as a small dose of mathematical suicide. By the end of their stay he was inordinately frustrated.

Not wasting time had become a veritable obsession, to the point where he obliterated from his life any activity that was not directly related to Goldbach's Conjecture – all except the two he couldn't reduce beyond a certain minimum, teaching and sleep. Yet he now got less sleep than he needed. Constant anxiety had brought insomnia with it, and this was aggravated by his excessive consumption of coffee, the fuel on which mathematicians run. With time, the constant preoccupation with the Conjecture made it impossible for him to relax. Falling or staying asleep became increasingly difficult and often he had to resort to sleeping pills. Occasional use gradually became steady and doses began to increase alarmingly, to the point of dependency, and this without the accompanying beneficial effect.

At about this time, a totally unexpected boost to his spirits came in the unlikely form of a dream. Despite his total disbelief in the supematural, Petros viewed it as prophetic, a definite omen straight from Mathematical Heaven.

It is not unusual for scientists totally immersed in a difficult problem to carry on their preoccupations into sleep; and although Petros was never honoured by nocturnal visitations from Ramanujan's Namakiri or any other revelatory deity (a fact that should not surprise us, considering his entrenched agnosticism), after the first year or so of his immersion in the Conjecture he began to have the occasional mathematical dream. In fact, his early visions of amorous bliss in the arms of 'dearest Isolde' became less frequent over time, giving their place to dreams of the Even Numbers, which appeared personified as couples of identical twins. They were involved in intricate, unearthly dumbshows, a chorus to the Primes, who were peculiar hermaphrodite, semi-human beings. Unlike the speechless Even Numbers, the Primes often chattered among themselves, usually in an unintelligible language, at the same time executing bizarre dance-steps. (By his admission, this dream choreography was most likely inspired by a production of Stravinsky's Rite of Spring that Petros had attended during his early years in Munich, when he still had time for such vanities.) On rare occasions the singular creatures spoke and then only in classical Greek – perhaps as a tribute to Euclid, who had awarded them infinitude. Even when their utterances made some linguistic sense, however, the content was mathematically either trivial or non-sensical. Petros specifically recalled one such: hapantes protoi perittoi, which means 'All prime numbers are odd', an obviously false Statement. (By a different reading of the word perittoi, however, it could also mean 'All prime numbers are useless', an interpretation which, interestingly, completely escaped my uncle's attention.)

Yet in a few rare instances there was something of substance in his dreams. He could deduce from the protagonists' sayings helpful hints that steered his research towards interesting, unexplored paths [10].

The dream that lifted his spirits came a few nights after he had proved his second important result. It was not directly mathematical, but laudatory, consisting of no more than a single image, a sparkling tableau vivant, but of such unearthly beauty! Leonard Euler was on the one side and Christian Goldbach (though he'd never seen a portrait, he immediately knew it to be him) on the other. The two men jointly held, from the sides, a golden wreath over the head of the central figure, which was none other than himself, Petros Papachristos. The triad was bathed in a nimbus of blinding light.

The dream's message could not be clearer: the proof of Goldbach's Conjecture would be ultimately his.

Spurred by the glorious spirit of this vision, his mood swung back to optimism and he coaxed himself onwards with added zest. Now, he should concentrate all his powers on his research. He could afford absolutely no distractions.

The painful gastrointestinal symptoms he had been having for some time (most of them by some strange coincidence occurring at times when they interfered with his university duties), a result of the constant, self-imposed pressure, gave him the pretext he needed. Armed with the opinion of a specialist, he went to see the Director of the School of Mathematics and requested a two-year, unpaid leave of absence.

The Director, an insignificant mathematician but a ferocious bureaucrat, was apparently waiting for an occasion to level with Professor Papachristos.

'I have read your doctor's recommendation, Herr Professor,’ he said in a sour tone. 'Apparently you suffer – like many in our School – from gastritis, a condition that is not exactly terminal. Isn't a two-year leave rather excessive?'

'Well, Herr Director,’ mumbled Petros, 'I also happen to be at a critical point in my research. While on my two-year leave I can complete it.'

The Director appeared genuinely surprised. 'Research? Oh, I had no idea! You see, the fact that you haven't published anything during all your years with us had led your colleagues to think that you were scientifically inactive.'

Petros knew the next question was inevitable:

'By the way, what exactly is it you are researching, Herr Professor?'

'We-ell,’ he replied meekly, 'I am investigating certain questions in Number Theory.'

The Director, an eminently practical man, considered Number Theory, a field notorious for the inapplicability of its results to the physical sciences, a complete waste of time. His own interest lay in differential equations and, years back, he had hoped that the addition of the inventor of the Papachristos Method to the faculty would perhaps put his own name on some joint publications. This, of course, had never come about.

'You mean Number Theory in general, Herr Professor?'

Petros suffered the ensuing cat-and-mouse game for a while, trying desperately to prevaricate concerning his real object. When, however, he realized he had not the slightest hope unless he convinced the Director of the importance of his work, he revealed the truth.

‘I’m working on Goldbach's Conjecture, Herr Director. But please don't tell anyone!'

The Director appeared startled. 'Oh? And how are you progressing?'

'Quite well, actually.'

'Which means you have arrived at some very interesting intermediate results. Am I right?'

Petros felt as if he were walking on a tightrope. How much could he safely reveal?

'Well… er…' He was fidgeting in his seat, sweating profusely. 'In fact, Herr Director, I believe I'm only one step away from the proof. If you would let me have my two years of unpaid leave, I will try to complete it.'

The Director knew Goldbach's Conjecture – who didn't? Despite the fact that it belonged to the cloud-cuckoo-land world of Number Theory it had the advantage of being an exceedingly famous problem. A success by Professor Papachristos (he was reputed to have, after all, a first-class mind) would definitely be to the great benefit of the university, the School of Mathematics and of course himself, its director. After pondering the matter for a while, he gave him a big smile and declared he wasn't unfavourable to the request.

When Petros went to thank him and say goodbye, the Director was all smiles.

'Good luck with the Conjecture, Herr Professor. I expect you back with great results!'

Having secured his two-year period of grace, he moved to the outskirts of Innsbruck, in the Austrian Tyrol, where he had rented a small cottage. As a forwarding address he left only the local poste restante. In his new, temporary abode he was a complete stranger. Here, he needn't fear even the minor distractions of Munich, a chance encounter with an acquaintance in the street or the solicitude of his housekeeper, whom he left behind to look after the empty apartment. His isolation would remain absolutely inviolate.

During his stay in Innsbruck, there was a development in Petros' life that turned out to have a beneficial effect both on his mood and, as a consequence, on his work: he discovered chess.

One evening, while out for his habitual walk, he stopped for a hot drink at a coffee-house, which happened to be the meeting-place of the local club. He had been taught the rules of chess and played a few games as a child, yet he remained to that day totally unaware of its profundity. Now, as he sipped his cocoa, his attention was caught by the game in progress at the next table and he followed it through with increasing interest. The next evening his footsteps led him to the same place, and the day after that as well. At first


through mere observation, he gradually began to grasp the fascinating logic of the game.

After a few visits, he accepted a challenge to play. He lost, which was an irritant to his antagonistic nature, particularly so when he learned that his opponent was a cattle-herder by occupation. He stayed up that night, recreating the moves in his mind, trying to pinpoint his mistakes. The next evenings he lost a few more games, but then he won one and felt immense joy, a feeling that spurred him on towards more victories.

Gradually, he became a habitue of the coffee-house and joined the chess club. One of the members told him about the huge volume of accumulated wisdom on the subject of the game's first moves, also known as 'opening theory'. Petros borrowed a basic book and bought the chess set that he was still using in his old age, at his house in Ekali. He'd always kept late nights, but in Innsbruck it wasn't due to Goldbach. With the pieces set out in front of him and the book in hand, he spent the hours before sleep teaching himself the basic openings, the 'Ruy Lopez', the 'King's' and 'Queen's Gambits', the 'Sicilian Defence'.

Armed with some theoretical knowledge he proceeded to win more and more often, to his huge satisfaction. Indeed, displaying the fanaticism of the recent convert, he went overboard for a while, spending time on the game which belonged to his mathematical research, going to the coffee-house earlier and earlier, even turning to his chessboard during the daylight hours to analyse the previous day's games. However, he soon disciplined himself and restricted his chess activity to his nightly outing and an hour or so of study (an opening, or a famous game) before bedtime. Despite this, by the time he left Innsbruck he was the undisputed local champion.

The change brought about in Petros' life by chess was considerable. From the moment he had first dedicated himself to proving Goldbach's Conjecture, almost a decade earlier, he had hardly ever relaxed from his work. However, for a mathematician to spend time away from the problem at hand is essential. Mentally to digest the work accomplished and process its results at an unconscious level, the mind needs leisure as well as exertion. Invigorating as the investigation of mathematical concepts can be to a calm intellect, it can become intolerable when the brain is overcome by weariness, exhausted by incessant effort.

Of the mathematicians of his acquaintance, each had his own way of relaxing. For Caratheodory it was his administrative duties at Berlin University. With his colleagues at the School of Mathematics it varied: for family men it was usually the family; for some it was sports; for some, collecting or the theatrical performances, concerts and other cultural events that were on constant offer at Munich. None of these, however, suited Petros – none engaged him sufficiently to provide distraction from his research. At some point he tried reading detective stories, but after he'd exhausted the exploits of the ultrarationalist Sherlock Holmes he found nothing eise to hold his attention. As for his long afternoon walks, they definitely did not count as relaxation. While his body moved, whether in the countryside or the city, by a serene lakeside or on a busy pavement, his mind was totally preoccupied with the Conjecture, the walking itself being no more than a way to focus on his research.

So, chess seemed to have been sent to him from heaven. Being by its nature a cerebral game, it has concentration as a necessary requirement. Unless matched with a much inferior opponent, and sometimes even then, the player's attention can only wander at a cost. Petros now immersed himself in the recorded encounters between the great players (Steinitz, Alekhine, Capablanca) with a concentration known to him only from his mathematical studies. While trying to defeat Innsbruck's better players he discovered that it was possible to take total leave of Goldbach, even if only for a few hours. Faced with a strong opponent he realized, to his utter amazement, that for a few hours he could think of nothing but chess. The effect was invigorating. The morning after a challenging game he would tackle the Conjecture with a clear and refreshed mind, new perspectives and connections emerging, just as he'd begun to fear that he was drying up.

The relaxing effect of chess also helped Petros to wean himself from sleeping pills. From then on, if some night he were overcome by fruitless anxiety connected with the Conjecture, his tired brain twisting and wandering in endless mathematical mazes, he would get up from bed, seat himself before the chess-board and go over the moves of an interesting game. Immersing himself in it, he would temporarily forget his mathematics, his eyelids would grow heavy and he would sleep like a baby in his armchair till morning.

Before his two years of unpaid leave were up, Petros took a momentous decision: he would publish his two important discoveries, the 'Papachristos Partition Theorem' and the other one.

This, it must be stressed, was not because he had now decided to be content with less. There was no defeatism whatsoever concerning his ultimate aim of proving Goldbach's Conjecture. In Innsbruck, Petros had calmly reviewed the state of knowledge on his problem. He'd gone over the results arrived at by other mathematicians before him and also he'd analysed the course of his own research. Retracing his steps and coolly assessing his achievement to date, two things became obvious: a) His two theorems on Partitions were important results in their own right, and b) They brought him no closer to the proof of the Conjecture – his initial plan of attack had not yielded results.

The intellectual peace he had achieved in Innsbruck resulted in a fundamental insight: the fallacy in his approach lay in the adoption of the analytic approach. He realized now that he had been led astray by the success of Hadamard and de la Vallee-Poussin in proving the Prime Number Theorem and also, especially, by Hardy's authority. In other words, he had been misled by the demands of mathematical fashion (oh yes, such a thing does exist!), demands that have no greater right to be considered Mathematical Truth than the annually changing whims of the gurus of haute couture do to be regarded as the Platonic Ideal of Beauty. The theorems arrived at through rigorous proof are indeed absolute and eternal, but the methods used to get to them are definitely not. They represent choices that are by definition circumstantial – which is why they change as often as they do.

Petros' powerful intuition now told him that the analytic method had all but exhausted itself. The time had come for something new or, to be exact, something old, a return to the ancient, time-honoured approach to the secrets of numbers. The weighty responsibility of redefining the course of Number Theory for the future, he now decided, lay on his shoulders: a proof of Goldbach's Conjecture using the elementary, algebraic techniques would settle the matter once and for all.

As to his two first results, the Partition Theorem and the other, they could now safely be released to the general mathematical population. Since they had been arrived at through the (no longer seemingly useful to him for proving the Conjecture) analytic method, their publication could not threaten unwelcome infringements on his future research.

When he returned to Munich, his housekeeper was delighted to see the Herr Professor in such good shape. She hardly recognized him, she said, he 'looked so robust, so flushed with good health'.

It was mid-summer and, unencumbered by academic obligations, he immediately started to compose the monograph that presented his two important theorems with their proofs. Seeing once again the harvest of his ten-year hard labours with the analytic method in concrete form, with a beginning, a middle and an end, complete and presented and explained in a structured way, Petros now felt deeply satisfied. He realized that, despite the fact that he had not yet managed to prove the Conjecture, he had done excellent mathematics. It was certain that the publication of his two theorems would secure him his first significant scientific laurels. (As already mentioned, he was indifferent to the lesser, applications-oriented interest in the 'Papachristos method for the solution of differential equations'.) He could now even allow himself some gratifying daydreams of what was in store for him. He could almost see the enthusiastic letters from colleagues, the congratulations at the School, the invitations to lecture on his discoveries at all the great universities. He could even envision receiving international honours and prizes. Why not – his theorems certainly deserved them!

With the beginning of the new academic year (and still working on the monograph) Petros resumed his teaching duties. He was surprised to discover that for the first time he was now enjoying his lectures. The required effort at clarification and explanation for the sake of his students increased his own enjoyment and understanding of the material he was teaching. The Director of the School of Mathematics was obviously satisfied, not only by the improved performance he was hearing about from assistants and students alike, but mainly by the information that Professor Papachristos was preparing a monograph for publication. The two years at Innsbruck had paid off. Even though his forthcoming work apparently did not contain the proof of Goldbach's Conjecture, it was already rumoured in the School that it put forward extremely important results.

The monograph was finished a little after Christmas and it came to about two hundred pages. It was titied, with the usual slightly hypocritical modesty of many mathematicians when publishing important results, 'Some Observations on the Problem of Partitions'. Petros had it typed at the School and mailed a copy to Hardy and Littlewood, purportedly asking them to go over it lest he had slipped into an undetected pitfall, lest some less-than-obvious deductive error had escaped him. In fact, he knew well that there were no pitfalls and no errors: he just relished the thought of the two paragons of Number Theory's surprise and amazement. In fact, he was already basking in their admiration for his achievement.

After he sent off the typescript, Petros decided he owed himself a small vacation before he turned once again full-time to his work on the Conjecture. He de-voted the next few days exclusively to chess.

He joined the best chess club in town, where he discovered to his delight that he could beat all but the very top players and give a hard time to the select few he could not easily overpower. He discovered a small bookshop owned by an enthusiast, where he bought weighty volumes of opening theory and collections of games. He installed the chessboard he'd bought at Innsbruck on a small table in front of his fireplace, next to a comfortable deep armchair upholstered in soft velvet. There he kept his nightly rendezvous with his new white and black friends.

This lasted for almost two weeks. 'Two very happy weeks,’ he told me, the happiness being made greater by the anticipation of Hardy's and Littlewood's doubtless enthusiastic response to the monograph.

Yet the response, when it arrived, was anything but enthusiastic and Petros' happiness was cut short. The reaction wasn't at all what he had anticipated. In a rather short note, Hardy informed him that his first important result, the one he'd privately christened the 'Papachristos Partitions Theorem', had been discovered two years before by a young Austrian mathematician. In fact, Hardy expressed his amazement that Petros had not been aware of this, since its publication had caused a sensation in the circles of number theorists and brought great acclaim to its young author. Surely he was following the developments in the field, or wasn't he? As for his second theorem: a rather more general version of it had been proposed without proof by Ramanujan in a letter to Hardy from India, a few days before his death in 1920, one of his last great intuitions. In the years since then, the Hardy-Littlewood partnership had managed to fill in the gaps and their proof had been published in the most recent issue of the Proceedings of the Royal Society, of which he included a copy.

Hardy concluded his letter on a personal note, expressing his sympathy to Petros for this turn of events. With it there was the suggestion, in the understated fashion of his race and class, that it might in the future be more profitable for him to stay in closer contact with his scientific colleagues. Had Petros been living the normal life of a research mathematician, Hardy pointed out, coming to the international congresses and colloquia, corresponding with his colleagues, finding out from them the progress of their research and letting them know of his, he wouldn't have come in second in both of these otherwise extremely important discoveries. If he continued in his self-imposed isolation, another such 'unfortunate occurrence' was bound to arise.

At this point in his narrative my uncle stopped. He had been talking for several hours. It was getting dark and the birdsong in the orchard had been gradually tapering off, a solitary cricket now rhythmically piercing the silence. Uncle Petros got up and moved with tired steps to turn on a lamp, a naked bulb that cast a weak light where we were seated. As he walked back towards me, moving slowly in and out of pale yellow light and violet darkness, he looked almost like a ghost.

'So that's the explanation,' I murmured, as he sat down.

'What explanation?' he asked absently.

I told him of Sammy Epstein and his failure to find any mention of the name Petros Papachristos in the bibliographical index for Number Theory, with the exception of the early joint publications with Hardy and Littlewood on the Riemann Zeta Function. I repeated the 'burnout theory' suggested to my friend by the 'distinguished professor' at our university: that his supposed occupation with Goldbach's Conjecture had been a fabrication to disguise his inactivity.

Uncle Petros laughed bitterly.

'Oh no! It was true enough, most favoured of nephews! You can tell your friend and his "distinguished professor" that I did indeed work on trying to prove Goldbach's Conjecture – and how and for how longl Yes, and I did get intermediate results – wonderful, important results – but I didn't publish them when I should have done and others got in there ahead of me. Unfortunately, in mathematics there's no silver medal. The first to announce and publish gets all the glory. There's nothing left for anyone eise.' He paused.

'As the saying goes, a bird in the hand is worth two in the bush and I, while pursuing the two, lost the one…'

Somehow I didn't think the resigned serenity with which he stated this conclusion was sincere.

'But, Uncle Petros,’ I asked him, 'weren't you horribly upset when you heard from Hardy?'

'Naturally I was – and "horribly" is exactly the word. I was desperate; I was overcome with anger and frustration and grief; I even briefly contemplated suicide. That was back then, however, another time, another seif. Now, assessing my life in retrospect, I don't regret anything I did, or did not do.'

'You don't? You mean you don't regret the opportunity you missed to become famous, to be acknowledged as a great mathematician?'

He lifted a warning finger. 'A very good mathematician perhaps, but not a great one! I had discovered two good theorems, that's all.'

"That's no mean achievement, surely!'

Uncle Petros shook his head. 'Success in life is to be measured by the goals you've set yourself. There are tens of thousands of new theorems published every year the world over, but no more than a handful per century that make history!'

'Still, Uncle, you yourself say your theorems were important.'

'Look at the young man,' he countered, 'the Austrian who published my – as I still think of it – Partitions Theorem before me: was he raised with this result to the pedestal of a Hubert, a Poincare? Of course not! Perhaps he managed to secure a small niche for his portrait, somewhere in a back room of the Edifice of Mathematics… but if he did, so what? Or, for that matter, take Hardy and Littlewood, top-class mathematicians both of them. They possibly made the Hall of Fame – a very large Hall of Farne, mind you – but even they did not get their statues erected at the grand entrance alongside Euclid, Archimedes, Newton, Euler, Gauss… That had been my only ambition and nothing short of the proof of Goldbach's Conjecture, which also meant cracking the deeper mystery of the primes, could possibly have lead me there…’ There was now a gleam in his eyes, a deep, focused intensity as he concluded: ‘I, Petros Papachristos, never having published anything of value, will go down in mathematical history – or rather will not go down in it – as having achieved nothing. This suits me fine, you know. I have no regrets. Mediocrity would never have satisfied me. To an ersatz, footnote kind of immortality, I prefer my flowers, my orchard, my chessboard, the conversation I'm having with you today. Total obscurity!'

With these words, my adolescent admiration for him as Ideal Romantic Hero was rekindled. But now it was marked by large doses of realism.

'So, Uncle, it was really a question of all or nothing, eh?'

He nodded slowly. 'You could put it that way, yes.'

'And was this the end of your creative life? Did you ever again work on Goldbach's Conjecture?'

He gave me a surprised look. 'Of course I did! In fact it was after that I did my most important work.' He smiled. 'We'll come to that by and by, dear boy. Don't worry, in my story there shall be no ignorabimus!’

Suddenly he laughed loudly at his own joke, too loudly for comfort, I thought. Then he leaned towards me and asked me in a low voice: 'Did you learn Gödel's Incompleteness Theorem?'

'I did,' I replied, 'but I don't see what it has to do with -'

He lifted his hand roughly, cutting me short.

' "Wir müssen wissen, wir werden wissen! In der Mathematik gibt es kein ignorabimus" ' he declaimed stridently, so loudly that his voice echoed against the pine trees and returned, to menace and haunt me. Sammy's theory of insanity instantly flashed through my mind. Could all this reminiscing have aggravated his condition? Could my uncle have finally become unhinged?

I was relieved when he continued in a more normal tone: '"We must know, we shall know! In mathematics there is no ignorabimus!” Thus spake the great David

Hubert in the International Congress, in 1900. A proclamation of mathematics as the heaven of Absolute Truth. The vision of Euclid, the vision of Consistency and Completeness…'

Uncle Petros resumed his story.

The vision of Euclid had been the transformation of a random collection of numerical and geometric observations into a well-articulated system, where one can proceed from the a priori accepted elementary truths and advance, applying logical operations, step by step, to rigorous proof of all true statements: mathematics as a tree with strong roots (the Axioms), a solid trunk (Rigorous Proof) and ever growing branches blooming with wondrous flowers (the Theorems). All later mathematicians, geometers, number theorists, algebraists, and more recently analysts, topologists, algebraic geometers, group-theorists, etc., the practitioners of all the new disciplines that keep emerging to this day (new branches of the same ancient tree) never veered from the great pioneer's course: Axioms-Rigorous Proof-Theorems.

With a bitter smile, Petros remembered the constant exhortation of Hardy to anyone (especially poor Ramanujan, whose mind produced them like grass on fertile soil) bothering him with hypotheses: 'Prove it! Prove it!' Indeed, Hardy liked saying, if a heraldic motto were needed for a noble family of mathematicians, there could be no better than Quod Erat Demonstrandum.

In 1900, during the Second International Congress of Mathematicians, held in Paris, Hubert announced that the time had come to extend the ancient dream to its ultimate consequences. Mathematicians now had at their disposal, as Euclid had not, the language of Formal Logic, which allowed them to examine, in a rigorous way, mathematics itself. The holy trinity of Axioms-Rigorous Proof-Theorems should hence be applied not only to the numbers, shapes or algebraic identities of the various mathematical theories but to the very theories themselves. Mathematicians could at last rigorously demonstrate what for two millennia had been their central, unquestioned credo, the core of the vision: that in mathematics every true statement is provable.

A few years later, Russell and Whitehead published their monumental Principia Mathematica, proposing for the first time a totally precise way of speaking about deduction, Proof Theory. Yet although this new tool brought with it great promise of a final answer to Hilbert's demand, the two English logicians fell short of actually demonstrating the critical property. The 'completeness of mathematical theories' (i.e. the fact that within them every true statement is provable) had not yet been proven, but there was now not the smallest doubt in anybody's mind or heart that one day, very soon, it would be. Mathematicians continued to believe, as Euclid had believed, that they dwelt in the Realm of Absolute Truth. The victorious cry emerging from the Paris Congress, 'We must know, we shall know, in Mathematics there is no ignorabimus,' still constituted the one unshakable article of faith of every working mathematician.

I interrupted this rather exalted historical excursion: 'I know all this, Uncle. Once you enjoined me to learn Gödel's theorem I obviously also had to find out about its background.'

'It's not the background,' he corrected me; 'it's the psychology. You have to understand the emotional climate in which mathematicians worked in those happy days, before Kurt Gödel. You asked me how I mustered up the courage to continue after my great disap-pointment. Well, here's how…'

Despite the fact that he hadn't yet managed to attain his goal and prove Goldbach's Conjecture, Uncle Petros firmly believed that his goal was attainable. Being himself Euclid's spiritual great-grandson, his trust in this was complete. Since the Conjecture was almost certainly valid (nobody with the exception of Ramanujan and his vague 'hunch' had ever seriously doubted this), the proof of it existed somewhere, in some form.

He continued with an example:

'Suppose a friend states that he has mislaid a key somewhere in his house and asks you to help him find it. If you believe his memory to be faultless and you have absolute trust in his integrity, what does it mean?'

'It means that he has indeed mislaid the key somewhere in his house.'

'And if he further ascertains that no one else entered the house since?'

'Then we can assume that it was not taken out of the house.'

'Ergo?'

'Ergo, the key is still there, and if we search long enough – the house being finite – sooner or later we will find it.'

My uncle applauded. 'Excellent! It is precisely this certainty that fuelled my optimism anew. After I had recovered from my first disappointment I got up one fine morning and said to myself: "What the hell – that proof is still out there, somewhere!"'

'And so?'

'And so, my boy, since the proof existed, one had but to find it!'

I wasn't following his reasoning.

'I don't see how this provided comfort, Uncle Petros: the fact that proof existed didn't in any way imply that you would be the one to discover it!'

He glared at me for not immediately seeing the obvious. 'Was there anyone in the whole wide world better equipped to do so than I, Petros Papachristos?'

The question was obviously rhetorical and so I didn't bother to answer it. But I was puzzled: the Petros Papachristos he was referring to was a different man from the self-effacing, withdrawn senior citizen I'd known since childhood.

Of course, it had taken him some time to recover from reading Hardy's letter and its disheartening news. Yet recover he eventually did. He pulled himself together and, his deposits of hope refilled through the belief in 'the existence of the proof somewhere out there', he resumed his quest, a slightly changed man. His misadventure, by exposing an element of vanity in his manic search, had created in him an inner core of peace, a sense of life continuing irrespective of Goldbach's Conjecture. His working schedule now became slightly more relaxed, his mind also aided by interludes of chess, more tranquil despite the constant effort.

In addition, the switch to the algebraic method, already decided in Innsbruck, made him feel once again the excitement of a fresh start, the exhilaration of entering virgin territory.

For a hundred years, from Riemann's paper in the mid-nineteenth Century, the dominant trend in Number Theory had been analytic. By now resorting to the ancient, elementary approach, my uncle was in the vanguard of an important regression, if I may be allowed the oxymoron. The historians of mathematics will do well to remember him for this, if for no other part of his work.

It must be stressed here that, in the context of Number Theory, the word 'elementary' can on no account be considered synonymous with 'simple' and even less so with 'easy'. Its techniques are those of Diophantus', Euclid's, Fermat's, Gauss's and Euler's great results and are elementary only in the sense of deriving from the elements of mathematics, the basic arithmetical operations and the methods of classical algebra on the real numbers. Despite the effectiveness of the analytic techniques, the elementary method stays closer to the fundamental properties of the integers and the results arrived at with it are, in an intuitive way clear to the mathematician, more profound.

Gossip had by now seeped out from Cambridge, that Petros Papachristos of Munich University had had a bit of bad luck, deferring publication of very important work. Fellow number theorists began to seek his opinions. He was invited to their meetings, which from that point on he would invariably attend, enlivening his monotonous lifestyle with occasional travel. The news had also leaked out (thanks here to

the Director of the School of Mathematics) that he was working on the notoriously difficult Conjecture of Goldbach, and that made his colleagues look on him with a mixture of awe and sympathy.

At an international meeting, about a year after his return to Munich, he ran across Littlewood. 'How's the work going on Goldbach, old chap?' he asked Perros.

'Always at it.'

'Is it true what I hear, that you're using algebraic methods?'

'It's true.'

Littlewood expressed his doubts and Petros surprised himself by talking freely about the content of his research. 'After all, Littlewood,’ he concluded, 'I know the problem better than anyone eise. My intuition tells me the truth expressed by the Conjecture is so fundamental that only an elementary approach can reveal it.'

Littlewood shrugged. 'I respect your intuition, Papachristos; it's just that you are totally isolated. Without a constant exchange of ideas, you may find yourself grappling with phantoms before you know it.'

'So what do you recommend,' Petros joked, 'issuing weekly reports of the progress of my research?'

'Listen,' said Littlewood seriously, 'you should find a few people whose judgement and integrity you trust. Start sharing; exchange, old chap!'

The more he thought about this suggestion, the more it made sense. Much to his surprise he realized that, far from frightening him, the prospect of discussing the progress of his work now filled him with pleasurable anticipation. Of course his audience would have to be small, very small indeed. If it was to consist of people 'whose judgement and integrity he trusted', that would of necessity mean an audience of no more than two: Hardy and Littlewood.

He started anew the correspondence with them that he'd interrupted a couple of years after he left Cambridge. Without stating it in so many words, he dropped hints about his intention to bring about a meeting during which he would present his work. Around Christmas of 1931, he received an official invitation to spend the next year at Trinity College. He knew that since, for all practical purposes, he had been absent from the mathematical world for a long long time, Hardy must have used all his influence to secure the offer. Gratitude, combined with the exciting prospect of a creative exchange with the two great number theorists, made him immediately accept.

Petros described his first few months in England, in the academic year 1932-33, as probably the happiest of his life. Memories of his first stay there, fifteen years earlier, infused his days at Cambridge with the enthusiasm of early youth, as yet untainted by the possibility of failure.

Soon after he arrived, he presented to Hardy and Littlewood the outline of his work to date with the algebraic method, and this gave him the first taste, after more than a decade, of the joy of peer recognition. It took him several mornings, standing at the blackboard in Hardy's office, to trace his progress in the three years since his volte-face from the analytic techniques. His two renowned colleagues, who were at first extremely sceptical, now began to see some advantages to his approach, Littlewood more so than Hardy.

'You must realize,’ the latter told him, 'that you're running a huge risk. If you don't manage to ride this approach to the end, you'll be left with precious little to show for it. Intermediate divisibility results, although quite charming, are not of much interest any more. Unless you can convince people that they can be useful in proving important theorems, like the Conjecture, they are not of themselves worth much.'

Petros was, as always, well aware of the risks he was taking.

'Still, something tells me you may well be on a good course,’ Littlewood encouraged him.

'Yes,' grumbled Hardy, 'but please do hurry up, Papachristos, before your mind begins to rot, the way mine's doing. Remember, at your age Ramanujan was already five years dead!'

This first presentation had taken place early in the Michaelmas term, yellow leaves falling outside the Gothic windows. During the winter months that followed, my uncle's work advanced more than it ever had. It was at this time that he also started using the method he called 'geometric'.

He began by representing all composite (i.e. non-prime) numbers by placing dots in a parallelogram, with the lowest prime divisor as width and the quotient of the number by it as height. For example, 15 is represented by 3 x 5 rows, 25 by 5 x 5,35 by 5 x 7 rows:

By this method, all even numbers are represented as double columns, as 2 x 2,2 x 3,2 x 4,2 x 5, etc. The primes, on the contrary, since they have no integer divisors, are represented as single rows, for example 5,7,11

Petros extended the insights from this elementary geometric analogy to arrive at number-theoretical conclusions.

After Christmas, he presented his first results. Since, however, instead of using pen and paper, he laid out his patterns on the floor of Hardy's study using beans, his new approach earned from Littlewood a teasing accolade. Although the younger man conceded that he found 'the famous Papachristos bean method' conceivably of some usefulness, Hardy was by now ourright annoyed.

'Beans indeed!' he said. 'There is a world of difference between elementary and infantile… Don't you forget it, Papachristos, this blasted Conjecture is difficult – if it weren't, Goldbach would have proved it himself!'

Petros, however, had faith in his intuition and attributed Hardy's reaction to the 'intellectual constipation brought about by age' (his words).

'The great truths in life are simple,' he told Littlewood later, when the two of them were having tea in his rooms. Littlewood countered him, mentioning the extremely complex proof of the Prime Number Theorem by Hadamard and de la Vallee-Poussin.

Then he made a proposal: 'What would you say to doing some real mathematics, old chap? I've been working for some time now on Hilbert's Tenth Problem, the solvability of Diophantine equations. I have this idea that I want to test, but I'm afraid I need help with the algebra. Do you think you could lend me a hand?'

Littlewood would have to seek his algebraic help elsewhere, however. Although his colleague's confidence in him was a boost to Petros' pride, he flatly declined. He was too exclusively involved with the Conjecture, he said, too deeply engrossed in it, to be able fruitfully to concern himself with anything eise.

His faith, backed by a stubborn intuition, in the 'infantile' (according to Hardy) geometric approach, was such that for the first time since he began work on the Conjecture, Petros now often had the feeling that he was almost a hair's breadth away from the proof. There were actually even a few exhilarating minutes, late on a sunny January afternoon, when he had the shortli ved illusion that he had succeeded – but, alas, a more sober examination located a small, but crucial mistake.

(I have to confess it, dear reader: at this point in my uncle's narrative I felt despite myself a quiver of vengeful joy. I remembered that summer in Pylos, a few years back, when I too had thought for a while I'd discovered the proof of Goldbach's Conjecture – although I did not then know it by name.)

His great optimism notwithstanding, Petros' occasional bouts of self-doubt, sometimes verging on despair (especially after Hardy's put-down of the geometric method), now became stronger than ever. Still, they could not curb his spirit. He fought them away by branding them the inevitable anguish preceding a great triumph, the onset of the labour pains leading into the delivery of the majestic discovery. After all, the night is darkest before dawn. He was, Petros felt certain, all but ready to run the final dash. One last concentrated burst of effort was all that was needed to award him the last brilliant insight.

Then, there would come the glorious finish

The heralding of Petros Papachristos' surrender, the termination of his efforts to prove Goldbach's Conjecture, came in a dream he had in Cambridge, sometime after Christmas – a portent whose full significance he did not at first fathom.

Like many mathematicians working for long periods with basic arithmetical problems, Petros had acquired the quality that has been called 'friendship with the integers', an extended knowledge of the idiosyncrasy, quirks and peculiarities of thousands of specific whole numbers. A few examples: a 'friend of the integers' will immediately recognize 199 or 457 or 1009 as primes. 220 he will automatically associate with 284 since they are linked by an unusual relationship (the sum of the integer divisors of each one is equal to the other). 256 he reads naturally as 2 to the eighth power, which he well knows to be followed by a number with great historical interest, since 257 can be expressed as 2^2^3+1, and a famous hypothesis held that all numbers of the form 2^2^n + 1 were prime. [11]

The first man my uncle had met who had this quality (and to the utmost degree) was Srinivasa Ramanujan. Petros had seen it demonstrated on many opportunities, and to me he recounted this anecdote:+ [+ Hardy also recounts the incident in his Mathematician's Apology without, however, acknowledging my uncle's presence.]

One day in 1918, he and Hardy were visiting him in the Sanatorium where he lay ill. To break the ice, Hardy mentioned that the taxi that had brought them had had the registration number 1729, which he personally found 'rather boring'. But Ramanujan, after pondering this for only a moment, disagreed vehemently: 'No, no, Hardy! It's a particularly interesting number – in fact, it's the smallest integer that can be expressed as the sum of two cubes in two different ways!" [12]

During the years that Petros worked on the Conjecture with the elementary approach, his own friendship with the integers developed to an extraordinary degree. Numbers ceased after a while being inanimate entities; they became to him almost alive, each with a distinct personality. In fact, together with the certainty that the solution existed somewhere out there, it added to his resolve to persevere during the most difficult of times: working with the integers, he felt, to quote him directly, 'constantly among friends'.

This familiarity caused an influx of specific numbers into his dreams. Out of the nameless, nondescript mass of integers that up until then crowded their nightly dramas, individual actors now began to emerge, even occasional protagonists. 65, for example, appeared for some reason as a City gentleman, with bowler hat and rolled umbrella, in constant companionship with one of his prime divisors, 13, a goblin-like creature, supple and lightning-quick. 333 was a fat slob, stealing bites of food from the mouths of its siblings 222 and 111, and 8191, a number known as a 'Mersenne Prime', invariably wore the attire of a French gamin, complete down to the Gauloise cigarette hanging from his lips.

Some of the visions were amusing and pleasant, others indifferent, still others repetitious and annoying. There was one category of arithmetical dream, however, which could only be called nightmarish, if not for horror or agony then for its profound, bottomless sadness. Particular even numbers would appear, personified as pairs of identical twins. (Remember that an even number is always of the form 2k, the sum of two equal integers). The twins would gaze on him fixedly, immobile and expressionless. But there was great, if mute, anguish in their eyes, the anguish of desperation. If they could have spoken, their words would have been: 'Come! Please. Hurry! Set us free!'

It was a variation on these sad apparitions that came to wake him one night late in January 1933. This was the dream that he termed in retrospect 'the herald of defeat'.

He dreamed of 2^100 (2 to the hundredth power, an enormous number) personified as two identical, freckled, beautiful dark-eyed girls, looking straight into his eyes. But now there wasn't just sadness in their look, as there had been in his previous visions of the Evens; there was anger, hatred even. After gazing at him for a long, long while (this in itself was sufficient cause to brand the dream a nightmare) one of the twins suddenly shook her head from side to side with jerky, abrupt movements. Then her mouth was contorted into a cruel smile, the cruelty being that of a rejected lover.

'You'll never get us,’ she hissed.

At this, Petras, drenched in sweat, jumped up from his bed. The words that 2^99 (that's one half of 2^I00) had spoken meant only one thing: He was not fated to prove the Conjecture. Of course, he was not a superstitious old woman who would give undue credence to omens. Yet the profound exhaustion of many fruitless years had now begun to take its toll. His nerves were not as strong as they used to be and the dream upset him inordinately.

Unable to go back to sleep, he went out to walk in the dark, foggy streets, to try to shake off its dreary feeling. As he walked in the first light among the ancient stone buildings, he suddenly heard fast foot-steps approaching behind him, and for a moment he was seized by panic and turned sharply round. A young man in athletic gear materialized out of the mist, running energetically, uttered a greeting and dis-appeared once again, his rhythmic breathing trailing off into complete silence.

Still upset by the nightmare, Petros wasn't sure whether this image had been real, or an overflow of his dream world. When, however, a few months later the very same young man came to his rooms at Trinity on a fateful mission, he instantly recognized him as the early-morning runner. After he was gone, he realized with hindsight that their first, dawn meeting had cryptically signalled the dark forewarning, coming as it did after the vision of 2^100, with its message of defeat.

The fatal meeting took place a few months after the first, early-morning encounter. In his diary Petros marks the exact date with a laconic comment – the first and last use of Christian reference I discovered in his diaries: '17 March 1933. Kurt Gödel's Theorem. May Mary, Mother of God, have mercy on me!'

It was late afternoon and he had been in his rooms all day, sitting forward in his armchair studying parallelograms of beans laid out on the floor before him, lost in thought, when there was a knock on the door.

'Professor Papachristos?'

A blond head appeared. Petros had a powerful visual memory and immediately recognized the young runner, who was full of excuses for disturbing him. 'Please forgive my barging in on you like this, Professor,’ he said, 'but I am desperate for your help.' Petros was quite surprised – he'd thought his presence at Cambridge had gone completely unremarked.

He wasn't famous, he wasn't even well known and, except at his almost nightly appearances at the university chess club, he hadn't exchanged two words with anyone other than Hardy and Littlewood during his stay.

'My help on what subject?'

'Oh, in deciphering a difficult German text – a mathematical text.' The young man apologized again for presuming to take up his time with such a lowly task. This particular article, however, was of such great importance to him that when he heard that a senior mathematician from Germany was at Trinity, he couldn't resist appealing to him for assistance in its precise translation.

There was something so childishly eager in his manner that Petros couldn't refuse him.

Td be glad to help you, if I can. What field is the article in?'

'Formal Logic, Professor. The Grundlagen, the Foundations of Mathematics.'

Petros felt a rush of relief that it wasn't in Number Theory – he'd feared for a moment the young caller might have wanted to pump him on his work on the Conjecrure, using help with the language merely as an excuse. As he was more or less finished with his day's work, he asked the young visitor to take a seat.

'What did you say your name was?'

'It's Alan Turing, Professor. I'm an undergraduate.'

Turing handed him the journal containing the article, opened at the right page.

'Ah, the Monatshefte für Mathematik und Physik,' said Petros, 'the Monthly Review for Mathematics and Physics, a highly esteemed publication. The title of the article is, I see, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme". In translation this would be… Let's see… "On the formally undecidable propositions of Principia Mathematica and similar Systems". The author is a Mr Kurt Gödel, from Vienna. Is he well known in this field?'

Turing looked at him surprised. 'You don't mean to say you haven't heard of this article, Professor?'

Petros smiled: 'My dear young man, mathematics too has been infected by the modern plague, overspecialization. I'm afraid I have no idea of what's being accomplished in Formal Logic, or any other field for that matter. Outside of Number Theory I am, alas, a complete innocent.'

'But Professor,' Turing protested, 'Gödel's Theorem is of interest to all mathematicians, and number theorists especially! Its first application is to the very basis of arithmetic, the Peano-Dedekind axiomatic system.'

To Turing's amazement, Petros also wasn't too clear about the Peano-Dedekind axiomatic system. Like most working mathematicians he considered Formal

Logic, the field whose main subject is mathematics itself, a preoccupation that was certainly over-fussy and quite possibly altogether urmecessary. Its tireless attempts at rigorous foundation and its endless examination of basic prindples he regarded, more or less, as a waste of time. The piece of popular wisdom, 'If it ain't broke, don't fix it,' could well define this attitude: a mathematician's job was to try to prove theorems, not perpetually ponder the status of their unspoken and unquestioned basis.

In spite of this, however, the passion with which his young visitor spoke had aroused Petros' curiosity. 'So, what did this young Mr Gödel prove, that is of such interest to number theorists?'

'He solved the Problem of Completeness,' Turing announced with stars in his eyes.

Petros smiled. The Problem of Completeness was nothing other than the quest for a formal demonstration of the fact that all true statements are ultimately provable.

'Oh, good,' Petros said politely. 'I have to tell you, however – no offence meant to Mr Gödel, of course – that to the active researcher, the completeness of mathematics has always been obvious. Still, it's nice to know that someone finally sat down and proved it.'

But Turing was vehemently shaking his head, his face flushed with excitement. "That's exactly the point, Professor Papachristos: Gödel did not prove it!'

Petros was puzzled. 'I don't understand, Mr Turing… You just said this young man solved the Problem of Completeness, didn't you?'

'Yes, Professor, but contrary to everybody's expectation – Hilbert's and Russell's included – he solved it in the negative! He proved that arithmetic and all mathematical theories are not complete!'

Petros was not familiar enough with the concepts of Formal Logic immediately to realize the full implications of these words. 'I beg your pardon?'

Turing knelt by his armchair, his finger stabbing excitedly at the arcane symbols filling Gödel's article. 'Here: this genius proved – conclusively proved! – that no matter what axioms you accept, a theory of numbers will of necessity contain unprovable propositions!'

'You mean, of course, the false propositions?'

'No, I mean true propositions – true yet impossible to prove!'

Petros jumped to his feet. 'This is not possible!'

'Oh yes it is, and the proof of it is right here, in these fifteen pages: "Truth is not always provable!"'

My uncle now felt a sudden dizziness overcome him. 'But… but this cannot be.'

He flipped hurriedly through the pages, striving to absorb in a single moment, if possible, the article's intricate argument, mumbling on, indifferent to the young man's presence.

'It is obscene… an abnormality… an aberration…'

Turing was smiling smugly. That's how all mathematicians react at first… But Russell and Whitehead have examined Gödel's proof and proclaimed it to be flawless. In fact, the term they used was "exquisite".'

Petros grimaced. '"Exquisite"? But what it proves – if it really proves it, which I refuse to believe – is the end of mathematics!’

For hours he pored over the brief but extremely dense text. He translated as Turing explained to him the underlying concepts of Formal Logic, with which he was unfamiliar. When they'd finished they took it again from the top, going over the proof step by step, Petros desperately seeking a faulty step in the deduction.

This was the beginning of the end.

It was past midnight when Turing left. Petros couldn't sleep. First thing the next morning he went to see Littlewood. To his great surprise, he already knew of Gödel's Incompleteness Theorem.

'How could you not have mentioned it even once?' Petros asked him. 'How could you know of the existence of something like that and be so calm about it?'

Littlewood didn't understand: 'What are you so upset about, old chap? Gödel is researching some very special cases; he's looking into paradoxes apparently inherent in all axiomatic systems. What does this have to do with us line-of-combat mathematicians?'

However, Petros was not so easily appeased. 'But, don't you see, Littlewood? From now on, we have to ask of every statement still unproved whether it can be a case of application of the Incompleteness Theorem… Every outstanding hypothesis or conjecture can be a priori undemonstrable! Hilbert's "in mathematics there is no ignorabimus" no longer applies; the very ground that we stood on has been pulled out from under our feet!'

Littlewood shrugged. 'I don't see the point of getting all worked up about the few unprovable truths, when there are billions of provable ones to tackle!'

'Yes, damn it, but how do we know which is which?'

Although Littlewood's calm reaction should have been comforting, a welcome note of optimism after the previous evening's disaster, it didn't provide Petros with a definite answer to the one and only, dizzying, terrifying question that had jumped into his mind the moment he'd heard of Gödel's result. The question was so horrible he hardly dared formulate it: what if the Incompleteness Theorem also applied to his problem? What if Goldbach's Conjecture was unprovable?

From Littlewood's rooms he went straight to Alan Turing, at his College, and asked him whether there had been any further progress in the matter of the Incompleteness Theorem, after Gödel's original paper. Turing didn't know. Apparently, there was only one person in the world who could answer his question.

Petros left a note to Hardy and Littlewood saying he had some urgent business in Munich and crossed the Channel that same evening. The next day he was in Vienna. He tracked his man down through an academic acquaintance. They spoke on the telephone and, since Petros didn't want to be seen at the university, they made an appointment to meet at the cafe of the Sacher Hotel.

Kurt Gödel arrived precisely on time, a thin young man of average height, with small myopic eyes behind thick glasses.

Petros didn't waste any time: "There is something I want to ask you, Herr Gödel, in strict confidentiality.'

Gödel, by nature uncomfortable at social intercourse, was now even more so. 'Is this a personal matter, Herr Professor?'

'It is professional, but as it refers to my personal research I would appreciate it – indeed, I would demand! – that it remain strictly between you and me. Please let me know, Herr Gödel: is there a procedure for determining whether your theorem applies to a given hypothesis?'

Gödel gave him the answer he'd feared. 'No.'

'So you cannot, in fact, a priori determine which statements are provable and which are not?'

'As far as I know, Professor, every unproved statement can in principle be unprovable.'

At this, Petros saw red. He felt the irresistible urge to grab the father of the Incompleteness Theorem by the scruff of the neck and bang bis head on the shining surface of the table. However, he restrained himself, leaned forward and clasped his arm tightly.

‘I’ve spent my whole life trying to prove Goldbach's Conjecture,’ he told him in a low, intense voice, 'and now you're telling me it may be unprovable?'

Gödel's already pale face was now totally drained of colour.

'In theory, yes -'

'Damn theory, man!' Petros' shout made the heads of the Sacher cafe's distinguished clientele turn in their direction. 'I need to be certain, don't you understand? I have a right to know whether I'm wasting my life!'

He was squeezing his arm so hard that Gödel grimaced in pain. Suddenly, Petros felt shame at the way he was carrying on. After all, the poor man wasn't personally responsible for the incompleteness of mathematics – all he had done was discover it! He released his arm, mumbling apologies.

Gödel was shaking. 'I un-understand how you fe-feel, Professor,’ he stammered, 'but I-I'm afraid that for the time being there is no way to answer yo-your question.'

From then on, the vague threat hinted at by Gödel's Incompleteness Theorem developed into a relentless anxiety that gradually came to shadow his every living moment and finally quench his fighting spirit.

This didn't happen overnight, of course. Petros persisted in his research for a few more years, but he was now a changed man. From that point on, when he worked he worked half-heartedly, but when he despaired his despair was total, so insufferable in fact that it took on the form of indifference, a much more bearable feeling.

'You see,' Petros explained to me, 'from the first moment I heard of it, the Incompleteness Theorem destroyed the certainty that had fuelled my efforts. It told me there was a definite probability I had been wandering inside a labyrinth whose exit I'd never find, even if I had a hundred lifetimes to give to the search. And this for a very simple reason: because it was possible that the exit didn't exist, that the labyrinth was an infinity of cul-de-sacs! O, most favoured of nephews, I began to believe that I had wasted my life chasing a chimera!'

He illustrated his new Situation by resorting once again to the example he'd given me earlier. The hypothetical friend who had enlisted his help in seeking a key mislaid in his house might (or again might not, but there was no way to know which) be suffering from amnesia. It was possible that the 'lost key' had never existed in the first place!

The comforting reassurance, on which his efforts of two decades had rested, had, from one moment to the next, ceased to apply, and frequent visitations of the Even Numbers increased his anxiety. Practically every night now they would return, injecting his dreams with evil portent. New images haunted his nightmares, constant variations on themes of failure and defeat. High walls were being erected between him and the Even Numbers, which were retreating in droves, farther and farther away, heads lowered, a sad, vanquished army receding into the darkness of desolate, wide, empty spaces… Yet, the worst of these visions, the one that never failed to wake him trembling and drenched in sweat, was of 2^100, the two freckled, dark-eyed, beautiful girls. They gazed at him mutely, their eyes brimming with tears, then slowly turned their heads away, again and again, their features being gradually consumed by darkness.

The dream's meaning was clear; its bleak symbolism did not need a soothsayer or a psychoanalyst to decipher it: alas, the Incompleteness Theorem applied to his problem. Goldbach's Conjecture was a priori unprovable.

Upon his return to Munich after the year in Cambridge, Petros resumed the external routine he had established before his departure: teaching, chess, and also a minimum of social life; since he now had nothing better to do, he began to accept the occasional invitation. It was the first time since his earliest childhood that preoccupation with mathematical truths didn't occupy the central role in his life. And although he did continue his research awhile, the old fervour was gone. From then on he spent no more than a few hours a day at it, working half-absently at his geometric method. He'd still wake up before dawn, go to his study and pace slowly up and down, picking his way among the parallelograms of beans laid out on the floor (he had pushed all the furniture against the walls to make room). He picked up a few here, added a few there, muttering absently to himself. This went on for a while and then, sooner or later, he drifted towards the armchair, sat, sighed and turned his attention to the chessboard.

This routine went on for another two or three years, the time spent daily at this erratic form of 'research' continuously decreasing to almost nil. Then, near the end of 1936, Petros received a telegram from Alan Turing, who was now at Princeton University:

I HAVE PROVED THE IMPOSSIBILITY OF A PRIORI DECIDABILITY STOP.

Exactly. stop. This meant, in effect, that it was impossible to know in advance whether a particular mathematical statement is provable: if it is eventually proven, then it obviously is – what Turing had managed to show was that as long as it remains unproven, there is absolutely no way of ascertaining whether its proof is impossible or simply very difficult.

The immediate corollary of this, which concerned Petros, was that if he chose to pursue the proof of Goldbach's Conjecture, he would be doing so at his own risk. If he continued with his research, it would have to be out of sheer optimism and positive fighting spirit. Of these two qualities, however – time, exhaustion, ill luck, Kurt Gödel and now Alan Turing assisting – he had run out.

STOP.

A few days after Turing's telegram (the date he gives in his diary is 7 December 1936) Petros informed his housekeeper that the beans would no longer be required. She swept them all up, gave them a good wash and turned them into a hearty cassoulet for the Herr Professor 's dinner.

Uncle Petros remained silent for a while, looking dejectedly at his hands. Beyond the small circle of pale yellow light around us, cast by the single light-bulb, there was now total darkness.

'So that's when you gave up?' I asked softly.

He nodded. 'Yes.'

'And you never again worked on Goldbach's Conjecture?'

'Never.'

'What about Isolde?'

My question seemed to startle him. 'Isolde? What about her?'

'I thought that it was to win her love you decided to prove the Conjecture – no?'

Uncle Petros smiled sadly.

'Isolde gave me "the beautiful journey", as our poet says. Without her I might "never have set out". [13] Yet, she was no more than the original stimulus. A few years after I had begun my work on the Conjecture her memory faded, she became no more than a phantasm, a bittersweet recollection… My ambitions became of a higher, more exalted variety.'

He sighed. 'Poor Isolde! She was killed during the Allied bombardment of Dresden, along with her two daughters. Her husband, the "dashing young lieutenant" for whom she'd abandoned me, had died earlier on the Eastern Front.'

The last part of my uncle's story had no particular mathematical interest:

In the years that followed history, not mathematics, became the determining force in his life. World events broke down the protective barrier which till then had kept him safe within the ivory tower of his research. In 1938, the Gestapo arrested his housekeeper and sent her to what was still in those days referred to as a 'work camp'. He didn't hire anybody to take her place, naively believing that she'd return soon, her arrest due to some 'misunderstanding'. (After the war's end he learned from a surviving relative that she'd died in 1943 in Dachau, just a short distance from Munich.) He started to eat out, returning home only to sleep. When he was not at the university he would hang out at the chess club, playing, watching or analysing games.

In 1939, the Director of the School of Mathematics, by then a prominent member of the Nazi party, indicated that Petros should immediately apply for German citizenship and formally become a subject of the Third Reich. He refused, not for any reasons of principle (Petros managed to go through life unhampered by any ideological burden) but because the last thing he wanted was to be involved once again with differential equations. Apparently, it was the Ministry of Defence that had suggested he apply for citizenship, with precisely this aim in mind. After his refusal he became in essence a persona non grata. In September 1940, a little before Italy's declaration of war on Greece would have made him an enemy alien subject to internment, he was fired from his post. After a friendly warning, he left Germany.

Having, by the strict criterion of published work, been mathematically inactive for more than twenty years, Petros was now academically unemployable and so he had to return to his homeland. During the first years of the country's occupation by the Axis powers he lived in the family house in central Athens, on Queen Sophia Avenue, with his recently widowed father and his newly-wed brother Anargyros (my parents had moved to their own house), devoting practically all his time to chess. Very soon, however, my newborn cousins with their cries and toddler activities became a much greater annoyance to him than the occupying Fascists and Nazis and he moved to the small, rarely used family cottage in Ekali.

After the Liberation, my grandfather managed to secure for Petros the offer of the Chair of Analysis at Athens University, through string-pulling and manoeuvring. He turned it down, however, using the spurious excuse that 'it would interfere with his research'. (In this instance, my friend Sammy's theory of Goldbach's Conjecture as my uncle's pretext for idleness proved completely correct.) Two years later, paterfamilias Papachristos died, leaving to his three sons equal shares of his business and the principal executive positions exclusively to my father and Anargyros. 'My eldest, Petros,’ his will expressly decreed, 'shall retain the privilege of pursuing his important mathematical research,’ i.e. the privilege of being supported by his brothers without doing any work.

'And after that?' I asked, still cherishing the hope that a surprise might be in store, an unexpected reversal on the last page.

'After that nothing,’ my uncle concluded. ‘For almost twenty years my life has been as you know it: chess and gardening, gardening and chess. Oh, and once a month a visit to the philanthropic Institution founded by your grandfather, to help them with the book-keeping. It's something towards the salvation of my soul, just in case there exists a hereafter.'

It was midnight by this time and I was exhausted. Still, I thought I should end the evening on a positive note and, after a big yawn and a stretch, I said: 'You are admirable, Uncle… if not for anything eise, for the courage and magnanimity with which you accepted failure.'

This comment, however, got a reaction of utter surprise. 'What are you talking about?' my uncle said. 'I didn't fail!'

Now the surprise was mine. 'You didn't?'

'Oh no, no, no, dear boy!' He shook his head from side to side. 'I see you didn't understand anything. I didn't fail – I was just unlucky!'

'Unlucky? You mean unlucky to have chosen such a difficult problem?'

'No,’ he said, now looking totally amazed at my inability to grasp an obvious point. 'Unlucky – that, by the way, is a mild word for it – to have chosen a problem that had no solution. Weren't you listening?' He sighed heavily. 'By and by, my suspicions were confirmed: Goldbach's Conjecture is unprovable!'

'But how can you be so sure about it?' I asked.

'Intuition,’ he answered with a shrug. 'It is the only tool left to the mathematidan in the absence of proof. For a truth to be so fundamental, so simple to state, and yet so unimaginably resistant to any form of systematic reasoning, there could have been no other explanation. Unbeknown to me I had undertaken a Sisyphean task.'

I frowned. 'I don't know about that,’ I said. 'But the way I see it -'

Now, however, Uncle Petros interrupted me with a laugh. 'You may be a bright boy,’ he said, but mathematically you are still no more than a foetus – whereas I, in my time, was a veritable, full-blown giant. So, don't go weighing your intuition against mine, most favoured of nephews!' Against that, of course, I couldn't argue.

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