'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'.