Uncle Petros and Goldbach's Conjecture

by Apostolos Doxiadis

  ‘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 yourself.’

  *

  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 82 plus 52 plus 32plus 12,’ he’d answer.

  ‘And 290?’

  ‘290 is equal to 122 plus 92 plus 72 plus 42.’

  ‘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 père 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 élite 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 naїve. 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 prime.’

  ‘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 firstborn.

  *

  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 Charlot-tenburg.

  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 Gold-bach 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 Tapachristos 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 mêlée of the Great War, arranged for him to settle for a while in neutral Switzerland. In Zurich, 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 blasé 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 V
allée-Poussin with it, had become tres à 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-Ra-manujan 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, Little-wood 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.