(I have to confess it, dear reader: at this point in my uncle's narrative I felt despite myself a quiver of vengeful joy. I remembered that summer in Pylos, a few years back, when I too had thought for a while I'd discovered the proof of Goldbach's Conjecture – although I did not then know it by name.)
His great optimism notwithstanding, Petros' occasional bouts of self-doubt, sometimes verging on despair (especially after Hardy's put-down of the geometric method), now became stronger than ever. Still, they could not curb his spirit. He fought them away by branding them the inevitable anguish preceding a great triumph, the onset of the labour pains leading into the delivery of the majestic discovery. After all, the night is darkest before dawn. He was, Petros felt certain, all but ready to run the final dash. One last concentrated burst of effort was all that was needed to award him the last brilliant insight.
Then, there would come the glorious finish
The heralding of Petros Papachristos' surrender, the termination of his efforts to prove Goldbach's Conjecture, came in a dream he had in Cambridge, sometime after Christmas – a portent whose full significance he did not at first fathom.
Like many mathematicians working for long periods with basic arithmetical problems, Petros had acquired the quality that has been called 'friendship with the integers', an extended knowledge of the idiosyncrasy, quirks and peculiarities of thousands of specific whole numbers. A few examples: a 'friend of the integers' will immediately recognize 199 or 457 or 1009 as primes. 220 he will automatically associate with 284 since they are linked by an unusual relationship (the sum of the integer divisors of each one is equal to the other). 256 he reads naturally as 2 to the eighth power, which he well knows to be followed by a number with great historical interest, since 257 can be expressed as 2^2^3+1, and a famous hypothesis held that all numbers of the form 2^2^n + 1 were prime. [11]
The first man my uncle had met who had this quality (and to the utmost degree) was Srinivasa Ramanujan. Petros had seen it demonstrated on many opportunities, and to me he recounted this anecdote:+ [+ Hardy also recounts the incident in his Mathematician's Apology without, however, acknowledging my uncle's presence.]
One day in 1918, he and Hardy were visiting him in the Sanatorium where he lay ill. To break the ice, Hardy mentioned that the taxi that had brought them had had the registration number 1729, which he personally found 'rather boring'. But Ramanujan, after pondering this for only a moment, disagreed vehemently: 'No, no, Hardy! It's a particularly interesting number – in fact, it's the smallest integer that can be expressed as the sum of two cubes in two different ways!" [12]
During the years that Petros worked on the Conjecture with the elementary approach, his own friendship with the integers developed to an extraordinary degree. Numbers ceased after a while being inanimate entities; they became to him almost alive, each with a distinct personality. In fact, together with the certainty that the solution existed somewhere out there, it added to his resolve to persevere during the most difficult of times: working with the integers, he felt, to quote him directly, 'constantly among friends'.
This familiarity caused an influx of specific numbers into his dreams. Out of the nameless, nondescript mass of integers that up until then crowded their nightly dramas, individual actors now began to emerge, even occasional protagonists. 65, for example, appeared for some reason as a City gentleman, with bowler hat and rolled umbrella, in constant companionship with one of his prime divisors, 13, a goblin-like creature, supple and lightning-quick. 333 was a fat slob, stealing bites of food from the mouths of its siblings 222 and 111, and 8191, a number known as a 'Mersenne Prime', invariably wore the attire of a French gamin, complete down to the Gauloise cigarette hanging from his lips.
Some of the visions were amusing and pleasant, others indifferent, still others repetitious and annoying. There was one category of arithmetical dream, however, which could only be called nightmarish, if not for horror or agony then for its profound, bottomless sadness. Particular even numbers would appear, personified as pairs of identical twins. (Remember that an even number is always of the form 2k, the sum of two equal integers). The twins would gaze on him fixedly, immobile and expressionless. But there was great, if mute, anguish in their eyes, the anguish of desperation. If they could have spoken, their words would have been: 'Come! Please. Hurry! Set us free!'
It was a variation on these sad apparitions that came to wake him one night late in January 1933. This was the dream that he termed in retrospect 'the herald of defeat'.
He dreamed of 2^100 (2 to the hundredth power, an enormous number) personified as two identical, freckled, beautiful dark-eyed girls, looking straight into his eyes. But now there wasn't just sadness in their look, as there had been in his previous visions of the Evens; there was anger, hatred even. After gazing at him for a long, long while (this in itself was sufficient cause to brand the dream a nightmare) one of the twins suddenly shook her head from side to side with jerky, abrupt movements. Then her mouth was contorted into a cruel smile, the cruelty being that of a rejected lover.
'You'll never get us,’ she hissed.
At this, Petras, drenched in sweat, jumped up from his bed. The words that 2^99 (that's one half of 2^I00) had spoken meant only one thing: He was not fated to prove the Conjecture. Of course, he was not a superstitious old woman who would give undue credence to omens. Yet the profound exhaustion of many fruitless years had now begun to take its toll. His nerves were not as strong as they used to be and the dream upset him inordinately.
Unable to go back to sleep, he went out to walk in the dark, foggy streets, to try to shake off its dreary feeling. As he walked in the first light among the ancient stone buildings, he suddenly heard fast foot-steps approaching behind him, and for a moment he was seized by panic and turned sharply round. A young man in athletic gear materialized out of the mist, running energetically, uttered a greeting and dis-appeared once again, his rhythmic breathing trailing off into complete silence.
Still upset by the nightmare, Petros wasn't sure whether this image had been real, or an overflow of his dream world. When, however, a few months later the very same young man came to his rooms at Trinity on a fateful mission, he instantly recognized him as the early-morning runner. After he was gone, he realized with hindsight that their first, dawn meeting had cryptically signalled the dark forewarning, coming as it did after the vision of 2^100, with its message of defeat.
The fatal meeting took place a few months after the first, early-morning encounter. In his diary Petros marks the exact date with a laconic comment – the first and last use of Christian reference I discovered in his diaries: '17 March 1933. Kurt Gödel's Theorem. May Mary, Mother of God, have mercy on me!'
It was late afternoon and he had been in his rooms all day, sitting forward in his armchair studying parallelograms of beans laid out on the floor before him, lost in thought, when there was a knock on the door.
'Professor Papachristos?'
A blond head appeared. Petros had a powerful visual memory and immediately recognized the young runner, who was full of excuses for disturbing him. 'Please forgive my barging in on you like this, Professor,’ he said, 'but I am desperate for your help.' Petros was quite surprised – he'd thought his presence at Cambridge had gone completely unremarked.
[11] It was Fermat who first stated the general form, obviously generalizing from age-old observations that this was true of the first four values of n, i.e. 2^2^1 +1 = 5, 2^2^2 +1 = 17, 2^2^3 +1 = 257, 2^2^4 + 1 = 65537, all prime. However, it was later shown that for n = 5, 2^2^5 +1 equals 4294967297, a number which is composite, since it's divisible by the primes 641 and 6700417. Conjectures are not always proved correct!