Uncle Petros and Goldbach's Conjecture

by Apostolos Doxiadis

  After the Liberation, my grandfather managed to secure for Petros the offer of the Chair of Analysis at Athens University, through string-pulling and manoeuvring. He turned it down, however, using the spurious excuse that ‘it would interfere with his research’. (In this instance, my friend Sammy’s theory of Goldbach’s Conjecture as my uncle’s pretext for idleness proved completely correct.) Two years later, paterfamilias Papachristos died, leaving to his three sons equal shares of his business and the principal executive positions exclusively to my father and Anargyros. ‘My eldest, Petros,’ his will expressly decreed, ‘shall retain the privilege of pursuing his important mathematical research,’ i.e. the privilege of being supported by his brothers without doing any work.

  ‘And after that?’ I asked, still cherishing the hope that a surprise might be in store, an unexpected reversal on the last page.

  ‘After that nothing,’ my uncle concluded. ‘For almost twenty years my life has been as you know it: chess and gardening, gardening and chess. Oh, and once a month a visit to the philanthropic institution founded by your grandfather, to help them with the book-keeping. It’s something towards the salvation of my soul, just in case there exists a hereafter.’

  It was midnight by this time and I was exhausted. Still, I thought I should end the evening on a positive note and, after a big yawn and a stretch, I said: ‘You are admirable, Uncle … if not for anything else, for the courage and magnanimity with which you accepted failure.’

  This comment, however, got a reaction of utter surprise. ‘What are you talking about?’ my uncle said. ‘I didn’t fail!’

  Now the surprise was mine. ‘You didn’t?’

  ‘Oh no, no, no, dear boy!’ He shook his head from side to side. ‘I see you didn’t understand anything. I didn’t fail — I was just unlucky!’

  ‘Unlucky? You mean unlucky to have chosen such a difficult problem?’

  ‘No,’ he said, now looking totally amazed at my inability to grasp an obvious point. ‘Unlucky — that, by the way, is a mild word for it — to have chosen a problem that had no solution. Weren’t you listening?’ He sighed heavily. ‘By and by, my suspicions were confirmed: Goldbach’s Conjecture is unprovable!’

  ‘But how can you be so sure about it?’ I asked.

  ‘Intuition,’ he answered with a shrug. ‘It is the only tool left to the mathematician in the absence of proof. For a truth to be so fundamental, so simple to state, and yet so unimaginably resistant to any form of systematic reasoning, there could have been no other explanation. Unbeknown to me I had undertaken a Sisyphean task.’

  I frowned. ‘I don’t know about that,’ I said. ‘But the way I see it—’

  Now, however, Uncle Petros interrupted me with a laugh. ‘You may be a bright boy,’ he said, ‘but mathematically you are still no more than a foetus — whereas I, in my time, was a veritable, full-blown giant. So, don’t go weighing your intuition against mine, most favoured of nephews!’

  Against that, of course, I couldn’t argue.

  * * *

  * The largest such pair known today is almost inconceivably enormous: 83533539014 +/-1.

  * 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.)

  * Numbers of the form a + bi, where a, b are real numbers and i is the ‘imaginary’ square root of -1.

  * This states that any odd number greater than 5 is the sum of three primes.

  * In his seminal work The Nature of Mathematical Discovery, Henri Poincaré demolishes the myth of the mathematician as a totally rational being. With examples drawn from history, as well as from his own research experience, he places special emphasis on the role of the unconscious in research. Often, he says, great discoveries happen unexpectedly, in a flash of revelation that comes in a moment of repose — of course, these can occur only to minds that are otherwise prepared through endless months or years of conscious work. It is in this aspect of the workings of a mathematician’s mind that revelatory dreams can play an important role, sometimes providing the route through which the unconscious announces its conclusions to the conscious mind.

  * 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. all prime. However, it was later shown that for equals 4,294,967,297, a number which is composite, since it’s divisible by the primes 641 and 6,700,417. Conjectures are not always proved correct!

  † Hardy also recounts the incident in his Mathematician’s Apology without, however, acknowledging my uncle’s presence.

  * Indeed, 1729 = 123 + 13 = 103 + 93, a property which does not apply for any smaller integer.

  * C. Cavafy, ‘Ithaca’.

  Three

  My first reaction to this extensive autobiographical account was one of admiration. Uncle Petros had given me the facts of his life with remarkable honesty. It wasn’t until a few days later, when the oppressive influence of his melancholy narrative diminished, that I realized everything he’d told me had been beside the point.

  Remember that our meeting had been initially arranged so that he could try to justify himself. His life’s story was only relevant to the extent that it explained his atrocious behaviour, assigning me in all my adolescent mathematical innocence the task of proving Goldbach’s Conjecture. Yet, during his long narrative he had not even touched on his cruel prank. He’d ranted on and on about his own failure (or maybe I should do him the favour of calling it ‘bad luck’), but about his decision to turn me away from studying mathematics and the method he had chosen to implement it, not a single word. Did he expect me automatically to draw the conclusion that his behaviour to me was determined by his own bitter life-experiences? It didn’t follow: although his life story was indeed a valid cautionary tale, it taught a future mathematician what pitfalls to avoid so as to make the most of his career — not how to terminate it.

  I let a few days go by before I went back to Ekali and asked him point-blank: could he now explain why he had attempted to dissuade me from following my inclination.

  Uncle Petros shrugged. ‘Do you want the truth?’

  ‘Of course, Uncle.’ I said. ‘What else?’

  ‘All right then. I believed from the first moment — and still do, I’m sorry to say — that you have no special gift for great mathematics.’

  I became, once again, furious. ‘Oh? And how on earth could you have known that? Did you ask me a single mathematical question? Did you ever set me a problem to solve, other than the unprovable, as you termed it, Conjecture of Christian Goldbach? I certainly hope you don’t have the nerve to tell me that you deduced my lack of mathematical ability from that!’

  He smiled, sadly. ‘You know the popular saying that the three conditions impossible to conceal are a cough, wealth and being in love? Well, to me there is a fourth: mathematical gift.’

  I laughed contemptuously. ‘Oh, and you can no doubt identify it at a glance, eh? Is it a look in the eye or a certain je ne sais quoi that betrays to your ultra-fine sensibility the presence of mathematical genius? Can you perhaps also determine one’s IQ with a handshake?’

  ‘Actually there is an element of that “look in the eye”,’ he replied, ignoring my sarcasm. ‘But in your case physiognomy was only a small part of it. The necessary — but not sufficient, mind you — precondition for supreme achievement is single-minded devotion. If you had the gift that you yourself would like to have had, dear boy, you wouldn’t have come asking for my blessing to study mathematics; you would have gone ahead and done it. That was the first tell-tale sign!’

  The more he explained himself, the angrier I got. ‘If you were so certain I wasn’t gifted, Uncle, why did you put me through the horrific experience of that summer? Why did I have to be sub
jected to the totally unnecessary humiliation of thinking myself a near-idiot?’

  ‘But, don’t you see?’ he answered merrily. ‘Goldbach’s Conjecture was my security! If by some remote chance I’d been wrong about you and, in the most unlikely instance, you were indeed earmarked for greatness, then the experience wouldn’t have crushed you. In fact it would not have been at all “horrific”, as you significantly termed it, but exciting and inspiring and invigorating. I gave you an ultimate trial of determination, you see: if, after failing to solve the problem I’d set you — as, of course, I knew you would — you came back eager to learn more, to persist in your attempt for better or for worse, then I’d see you might have it in you to become a mathematician. But you … you weren’t even curious enough to ask the solution! Indeed, you even gave me a signed declaration of your incompetence!’

  The pent-up anger of many years now exploded. ‘Do you know something, you old bastard? You may once have been a good mathematician, but as a human being you rate zero! Absolutely, totally zilch!’

  To my surprise, this opinion was rewarded with a huge, hearty smile. ‘On that, most favoured of nephews, I couldn’t agree with you more!’

  A month later I returned to the United States to prepare for my Senior year. I now had a new room-mate, unrelated to mathematics. Sammy had meanwhile graduated and was at Princeton, already deeply involved in the problem that would in due course become his doctoral dissertation — with the exotic title: ‘The orders of the torsion subgroups of Ωn and the Adams spectral sequence’.

  On my first free weekend I took the train and went to visit him. I found him quite changed, much more nervous and irritable than I had known him in the year of our association. He’d also acquired some kind of facial tic. Obviously, the torsion subgroups of Ωn (whatever they were) had taken their toll on his nerves. We had dinner at a small pizza place across from the university and there I gave him a shortened version of Uncle Petros’ story, as I’d heard it from him. He listened without once interrupting for question or comment.

  After I was finished, he summed up his reaction in two words: ‘Sour grapes.’

  ‘What?’

  ‘You should know — Aesop was a Greek.’

  ‘What’s Aesop got to do with it?’

  ‘Everything. The fable of the fox who couldn’t reach a yummy bunch of grapes and therefore decided they were unripe anyway. What a wonderful excuse your uncle found for his failure: he put the blame on Kurt Gödel! Wow!’ Sammy burst out laughing. ‘Audacious! Unheard of! But I have to grant it to him, it is original; in fact it’s unique, it should go into some book of records! Never before has there been a mathematician seriously attributing his failure to find a proof to the Incompleteness Theorem!’

  Although Sammy’s words echoed my own first doubts, I lacked the mathematical knowledge to understand this immediate verdict.

  ‘So, you think it’s impossible that Goldbach’s Conjecture is unprovable?’

  ‘Man, what does “impossible” mean in this context?’ Sammy sneered. ‘As your uncle correctly told you, there is, thanks to Turing, no way of telling with certainty that a statement is a priori unprovable. But if mathematicians involved in advanced research started invoking Gödel, no one would ever go near the interesting problems — you see, in mathematics the interesting is always difficult. The Riemann Hypothesis has not yielded to proof after more than a century? A case of application of the Incompleteness Theorem! The Four Colour Problem? Likewise! Fer-mat’s Last Theorem still unproved? Blame it on evil Kurt Gödel! No one would ever have touched Hilbert’s Twenty-three Problems;* indeed it’s conceivable that all mathematical research, except the most trivial, would come to an end. Abandoning the study of a particular problem because it might be unprovable is like … like …’ His face lit up when he found the appropriate analogy. ‘Why, it’s like not going out in the street for fear that a brick might fall on your head and kill you!’

  ‘Let’s face it,’ he concluded, ‘your Uncle Petros simply and plainly failed to prove Goldbach’s Conjecture, like many greater men before him. But because, unlike them, he had spent his whole creative life on the problem, admitting his failure was unbearable. So, he concocted for himself this far-fetched, extravagant justification.’

  Sammy raised his soda-glass in a mock toast. ‘Here’s to far-fetched excuses,’ he said. Then he added in a more serious tone: ‘Obviously, for Hardy and Littlewood to have accepted him as a collaborator, your uncle must have been a gifted mathematician. He could have made a great success of his life. Instead, he wilfully chose to throw it away by setting himself an unattainable goal and going after a notoriously difficult problem. His sin was Pride: he presumed that he would succeed where Euler and Gauss had failed.’

  I was laughing now.

  ‘What’s so funny?’ asked Sammy.

  ‘After all these years of grappling with the mystery of Uncle Petros,’ I said, ‘I’m back to square one. You just repeated my father’s words, which I high-handedly rejected as philistine and coarse in my adolescence: “The secret of life, my son, is to set yourself attainable goals.” It’s exactly what you are saying now. That he didn’t do so is, indeed, the essence of Petros’ tragedy!’

  Sammy nodded. ‘Appearances are after all deceptive,’ he said with mock-solemnity. ‘It turns out the wise elder in the Papachristos family is not your Uncle Petros!’

  I slept on the floor of Sammy’s room that night, to the familiar sound of his pen scratching on paper accompanied by the occasional sigh or groan, as he struggled to untangle himself from the knots of a difficult topological problem. He left early in the morning to attend a seminar and in the afternoon we met at the Mathematics Library at Fine Hall, as arranged.

  ‘We are going sightseeing,’ he said. ‘I have a surprise for you.’

  We walked a distance on a long suburban road lined with trees and strewn with yellow leaves.

  ‘What courses are you taking this year?’ Sammy asked as we walked towards our mysterious destination.

  I started to list them: Introduction to Algebraic Geometry, Advanced Complex Analysis, Group Representation Theory …

  ‘What about Number Theory?’ he interrupted.

  ‘No. Why do you ask?’

  ‘Oh, I’ve been thinking about this business with your uncle. I wouldn’t want you getting any crazy ideas into your head about following family tradition and tackling —’

  I laughed. ‘Goldbach’s Conjecture? Not bloody likely!’

  Sammy nodded. That’s good. Because I have a suspicion that you Greeks are attracted to impossible problems.’

  ‘Why? Do you know any others?’

  ‘A famous topologist here, Professor Papakyri-akopoulos. He’s been struggling for years on end to prove the “Poincaré Conjecture” — it’s the most famous problem in low-dimensional topology, unproved for more than sixty years … ultra-hyper-difficult!’

  I shook my head. ‘I wouldn’t touch anybody’s famous unproved ultra-hyper-difficult problem with a ten-foot pole,’ I assured him.

  ‘I’m relieved to hear it,’ he said.

  We had reached a large nondescript building with extensive grounds. Once we had entered, Sammy lowered his voice.

  ‘I got a special permit to come, in your honour,’ he said.

  ‘What is this place?’

  ‘You’ll see.’

  We walked down a corridor and entered a large, darkish room, with the atmosphere of a slightly shabby but genteel English gentlemen’s club. About fifteen men, ranging from middle-aged to elderly, were seated in leather armchairs and couches, some by the windows, reading newspapers in the scanty daylight, others talking in little groups.

  We settled ourselves at a little table in a corner.

  ‘See that guy over there?’ Sammy said in a low voice, pointing to an old Asian gentleman, quietly stirring his coffee.

  ‘Yes?’

  ‘He is a Nobel Prize in Physics. And that other one at the far end’ — he indica
ted a plump, red-haired man gesturing heatedly as he spoke to his neighbour with a strong accent — ‘is a Nobel Prize in Chemistry.’ Then he directed my attention to two middle-aged men seated at a table near us. ‘The one on the left is André Weil —’

  ‘The André Weil?’

  ‘Indeed, one of the greatest living mathematicians. And the other one with the pipe is Robert Oppen-heimer — yes, the Robert Oppenheimer, the father of the atom bomb. He’s the Director.’

  ‘Director of what?’

  ‘Of this place here. You are now in the Institute for Advanced Study, think-tank for the world’s greatest scientific minds!’

  I was about to ask more when Sammy cut me short. ‘Shh! Look! Over there!’

  A most odd-looking man had just come in through the door. He was about sixty, of average height and emaciated to an extreme degree, wearing a heavy overcoat and a knitted cap pulled down over his ears. He stood for a moment and peered at the room vaguely through extremely thick glasses. No one paid him any attention: he was obviously a regular. He made his way slowly to the tea and coffee table without greeting anybody, filled a cup with plain boiling water from the kettle and made his way to a seat by a window. He slowly removed his heavy overcoat. Underneath it he was wearing a thick jacket over at least four or five layers of sweaters, visible through his collar.

  ‘Who is that man?’ I whispered.

  ‘Take a guess!’