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 ideaclass="underline" 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?
[6] The largest such pair known today is almost inconceivably enormous: 835335^39014 +/-1.
[7] Let k be a given integer. The set (k + 2)! + 2, (k + 2)! + 3, (k + 2)! + 4… (k + 2)! + (k + 1), (k + 2)! + (k + 2) contains k integers none of which is prime, since each is divisible by 2,3,4…, k + 1, k + 2 respectively. (The symbol k!, also known as 'k factorial', means the product of all the integers from 1 to k.)