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 proposaclass="underline" '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.