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.