Uncle Petros and Goldbach

by Apostolos Doxiadis

  If he was telling me all this in order to discourage me, he had chosen the wrong route.

  'That's what I'm after too, Uncle Petros,’ I responded excitedly. 'I don't want to be an engineer; I don't want to work in the family business. I want to immerse myself in real mathematics, just like you… just like Goldbach's Conjecture!'

  I'd blown it! Before I'd left for Ekali I had decided that any reference to the Conjecture should be avoided like the devil during our conversation. But in my carelessness and excitement I'd let it slip out.

  Although Uncle Petros remained expressionless, I noticed a slight tremor run down his hand.' Who's spoken to you about Goldbach's Conjecture?' he asked quietly.

  'My father,' I murmured.

  'And what did he say, precisely?'

  'That you tried to prove it.'

  'Just that?'

  'And… and that you didn't succeed.'

  His hand was steady again. 'Nothing else?'

  'Nothing else.'

  'Hm,’ he said. 'Suppose we make a deal?'

  'What sort of a deal?'

  'Listen to me: the way I see things, in mathematics as in the arts – or in sports, for that matter – if you're not the best, you're nothing. A civil engineer, or a lawyer, or a dentist who is merely capable may yet lead a creative and fulfilling professional life. However, a mathematician who is just average – I'm talking about a researcher, of course, not a high-school teacher – is a living, walking tragedy…'

  'But Uncle,’ I interrupted, 'I haven't the slightest intention of being "just average". I want to be Number One!'

  He smiled. 'In that at least you definitely resemble me. I too was overambitious. But you see, dear boy, good intentions are, alas, not enough. This is not like many other fields where diligence always pays. To get to the top in mathematics you also need something more, the absolutely necessary condition for success.'

  'Which one is that?'

  He gave me a puzzled look, for ignoring the obvious.

  'Why, the talent! The natural predisposition in its more extreme manifestation. Never forget it: Mathematicus nascitur, non fit – A mathematician is born, not made. If you don't carry the special aptitude in your genes, you will labour in vain all your life and one day you will end up a mediocrity. A golden mediocrity, perhaps, but a mediocrity nevertheless!'

  I looked him straight in the eye.

  'What's your deal, Uncle?'

  He hesitated for a moment, as if thinking it over. Then he said: 'I don't want to see you following a course that will lead to failure and unhappiness. Therefore I'm proposing that you will make a binding promise to me to become a mathematician if and only if you're supremely gifted. Do you accept?'

  I was disconcerted. 'But how on earth can I determine that, Uncle?'

  'You can't and you don't need to,' he said with a sly little smile. 'I will.'

  'You?'

  'Yes. I will set you a problem, which you will take home with you and attempt to solve. By your success, or failure, I will measure your potential for mathematical greatness with great accuracy.'

  I had mixed feelings for the proposed deal: I hated tests but adored challenges.

  'How much time will I have?' I asked. Uncle Petros half-closed his eyes, considering this. 'Mmm… Let's say till the beginning of school, the first of October. That gives you almost three months.'

  Ignorant as I was, I believed that in three months I could solve not one but any number of mathematical problems. 'That much!'

  'Well, the problem will be difficult,’ he pointed out. 'It's not one just anybody can solve, but if you've got what it takes to become a great mathematician, you will manage. Of course, you will swear that you will seek help from no one and you will not consult any books.' 'I swear,’ I said. He fixed his stare on me. 'Does that mean you accept the deal?'

  I heaved a deep sigh. 'I do!'

  Without a word, Uncle Petros disappeared briefly and returned with paper and pencil. He now became businesslike, mathematician to mathematician.

  'Here's the problem… I assume you already know what a prime number is?'

  'Of course I know, Uncle! A prime is an integer greater than 1 that has no divisors other than itself and unity. For example 2,3,5,7,11,13, and so on.'

  He appeared pleased with the precision of my definition. 'Wonderful! Now tell me, please, how many prime numbers are there?'

  I suddenly feit out of my depth. 'How many?'

  'Yes, how many. Haven't they taught you that at school?'

  'No.'

  My uncle sighed a deep sigh of disappointment at the low quality of modern Greek mathematical education.

  'All right, I will tell you this because you will need it: the primes are infinite, a fact first proven by Euclid in the third century BC. His proof is a gem of beauty and simplicity. By using reductio ad absurdum, he first assumes the contrary of what he wants to prove, namely that the primes are finite. So…'

  With fast vigorous jabs at the paper and a few explanatory words Uncle Petros laid out for my benefit our wise ancestor's proof, also giving me my first example of real mathematics.

  '… which, however,’ he concluded, 'is contrary to our initial assumption. Assuming finiteness leads to a contradiction; ergo the primes are infinite. Quod erat demonstrandum.'

  'That's fantastic, Uncle,’ I said, exhilarated by the ingeniousness of the proof. 'It's so simple!'

  'Yes,’ he sighed, 'so simple, yet no one had thought of it before Euclid. Consider the lesson behind this: sometimes things appear simple only in retrospect.'

  I was in no mood for philosophizing. 'Go on now, Uncle. State the problem I have to solve!'

  First he wrote it out on a piece of paper and then he read it to me.

  'I want you to try to demonstrate,’ he said, 'that every even number greater than 2 is the sum of two primes.'

  I considered it for a moment, fervently praying for a flash of inspiration that would blow him away with an instant solution. As it wasn't forthcoming, however, I just said:'That's all?'

  Uncle Petros wagged his finger in warning. 'Ah, it's not that simple! For every particular case you can consider, 4 = 2 +1,6 = 3 + 3,8 = 3 + 5,10 = 3 + 7,12 = 7 + 5, 14 = 7 + 7, etc., it's obvious, although the bigger the numbers get the more extensive the calculating. However, since there is an infinity of evens, a case-by-case approach is not possible. You have to find a general demonstration and this, I suspect, you may find more difficult than you think.'

  I got up. 'Difficult or not,’ I said, 'I will do it! I'm going to start work right away.'

  As I was on my way to the gate he called from the kitchen window. 'Hey! Aren't you going to take the paper with the problem?'

  A cold wind was blowing and I breathed in the exhalation of the moist soil. I don't think that ever in my life, whether before or after that brief moment, have I felt so happy, so full of promise and anticipation and glorious hope.

  'I don't need to, Uncle,’ I called back. 'I remember it perfectly: Every even number greater than 2 is the sum of two primes. See you on October the first with the solution!'

  His stern reminder found me in the street: 'Don't forget our deal,' he shouted. 'Only if you solve the problem can you become a mathematician!'

  A rough summer lay in store for me.

  Luckily, my parents always packed me off to my maternal uncle's house in Pylos for the hot months, July and August. That meant that, removed from my father's range, at least I didn't have the additional problem (as if the one Uncle Petros had set me were not enough) of having to conduct my work in secret. As soon as I arrived in Pylos I spread out my papers on the dining-room table (we always ate outdoors in the summer) and declared to my cousins that until further notice I would not be available for swimming, games and visits to the open-air movie theatre. I began to work at the problem from morning to night, with minimal interruption.

  My aunt fussed in her good-natured manner: 'You're workirvg too much, dear boy. Take it easy. It's summer vacation. Leav
e the books aside for a while. You came here to rest.'

  I, however, was determined not to rest until final victory. I slaved at my table incessantly, scribbling away on sheet after sheet of paper, approaching the problem from this side and that. Often, when I felt too exhausted for abstract deductive reasoning, I would test specific cases, lest Uncle Petros had set me a trap by asking me to demonstrate something obviously false. After countless divisions I had created a table of the first few hundred primes (a primitive, self-made Eratosthenes' Sieve [1]) which I then proceeded to add, in all possible pairs, to confirm that the principle indeed applied. In vain did I search for an even number within this boundary that didn't fit the required condition – all of them turned out to be expressible as the sum of two primes.

  At some point in mid-August, after a succession of late nights and countless Greek coffees, I thought for a few happy hours that I'd got it, that I'd found the solution. I filled several pages with my reasoning and mailed them, by special delivery, to Uncle Petros. I had barely enjoyed my triumph for a few days when the postman brought me the telegram:

  THE ONLY THING YOU HAVE DEMONSTRATED IS THAT EVERY EVEN NUMBER CAN BE EXPRESSED AS THE SUM OF ONE PRIME AND ONE ODD WHICH HOWEVER IS OBVIOUS STOP

  It took me a week to recover from the failure of my first attempt and the blow to my pride. But recover I did and half-heartedly I resumed work, this time employing the redudio ad absurdum:

  'Let us assume there is an even number n which cannot be expressed as the sum of two primes. Then…'

  The longer I laboured on the problem the more apparent it became that it expressed a fundamental truth regarding the integers, the materia prima of the mathematical universe. Soon I was driven to wondering about the precise way in which the primes are distributed among the other integers or the procedure which, given a certain prime, leads us to the next. I knew that this Information, were I to possess it, would be extremely useful in my plight and once or twice I was tempted to search for it in a book. However, loyal to my commitment not to seek outside help, I never did.

  By stating Euclid's demonstration of the infinity of the primes, Uncle Petros said he'd given me the only tool I needed to find the proof. Yet I was making no progress.

  At the end of September, a few days before the beginning of my last year in school, I found myself once again in Ekali, morose and crestfallen. Since Uncle Petros didn't have a telephone, I had to go through with this in person.

  'Well?' he asked me as soon as we sat down, after I'd stiffly refused his offer of a sour-cherry drink. 'Did you solve the problem?'

  'No,' I said. 'As a matter of fact, I didn't.'

  The last thing I wanted at that point was to have to trace the course of my failure or have him analyse it for my sake. What's more, I had absolutely no curiosity to learn the solution, the proof of the principle. All I wished was to forget everything even remotely related to numbers, whether odd or even – not to mention prime.

  But Uncle Petros wasn't willing to let me off easily. 'That's that then,’ he said. 'You remember our deal, don't you?'

  I found his need officially to ratify his victory (as, for some reason, I was certain he viewed my defeat) intensely annoying. Yet I wasn't planning to make it sweeter for him by displaying any hint of hurt feelings.

  'Of course I do, Uncle, as I'm sure you do too. Our deal was that I wouldn't become a mathematician unless I solved the problem -'

  'No!' he cut me off, with sudden vehemence. 'The deal was that unless you solved the problem you'd make a binding promise not to become a mathematician!'

  I scowled at him. 'Precisely,’ I agreed. 'And as I haven't solved the problem -'

  'You will now make a binding promise,' he interrupted, a second time completing the sentence, stressing the words as if his life (or mine, rather) depended on it.

  'Sure,’ I said, forcing myself to sound nonchalant, 'if it pleases you, I'll make a binding promise.'

  His voice became harsh, cruel even. 'It's not a question of pleasing me, young man, but of honouring our agreement! You will pledge to stay away from mathematics!'

  My annoyance instantly developed into full-fledged hatred.

  'All right, Uncle,' I said coldly. 'I pledge to stay away from mathematics. Happy now?'

  But as I got up to go he lifted his hand, menacingly. 'Not so fast!'

  With a quick move he got a sheet of paper out of his pocket, unfolded it and stuck it in front of my nose. This was it:

  /, the undersigned, being in full possession of my faculties, hereby solemnly pledge that, having failed in my examination for a higher mathematical capability and in accordance with the agreement made with my uncle, Petros Papachristos, I will never work towards a mathematics degree at an Institution of higher learning, nor in any other way attempt to pursue a professional career in mathematics.

  I stared at him in disbelief.

  'Sign!' he commanded.

  'What's the use of this?' I growled, now making no effort to conceal my feelings.

  'Sign,’ he repeated unmoved. 'A deal is a deal!'

  I left his extended hand holding the fountain pen suspended in mid-air, got out my ballpoint and jabbed in my signature. Before he had time to say anything more I threw the paper at him and made a wild rush to the gate.

  'Wait!' he shouted, but I was already outside.

  I ran and ran and ran until I was safely out of his hearing and then I stopped and, still breathless, broke down and cried like a baby, tears of anger and frustration and humiliation streaming down my face.

  I neither saw nor spoke to Uncle Petros during my last year of school, and in the following June I made up an excuse to my father and stayed home during the traditional family visit to Ekali.

  My experience of the previous summer had had the exact result that Uncle Petros had, doubtless, intended and foreseen. Irrespective of any obligation to keep my part of our 'deal', I had lost all desire to become a mathematician. Luckily, the side effects of my failure were not extreme, my rejection was not total and my superior performance at school continued. As a consequence, I was admitted to one of the best universities in the United States. Upon registration I declared a major in Economics, a choice I abided by till my Junior year [2]. Apart from the basic requirements, Elementary Calculus and Linear Algebra (incidentally, I got As in both), I took no other mathematics courses in my first two years.

  Uncle Petros' successful (at first, anyway) ploy had been based on the application of the absolute determinism of mathematics to my life. He had taken a risk, of course, but it was a well-calculated one: the possibility of my discovering the identity of the problem he had assigned me in the course of elementary university mathematics was minimal. The field to which it belongs is Number Theory, only taught in electives aimed at mathematics majors. Therefore it was reasonable for him to assume that, as long as I kept my pledge, I would complete my university studies (and conceivably my life) without learning the truth.

  Reality, however, is not as dependable as mathematics, and things turned out differently.

  On the first day of my Junior year I was informed that Fate (for who else can arrange coincidences such as this?) had assigned that I share my dormitory room with Sammy Epstein, a slightly built boy from Brooklyn renowned among undergraduates as a phenomenal maths prodigy. Sammy would be getting his degree that same year at the age of seventeen and, although he was nominally still an undergraduate, all his classes were already at advanced graduate level. In fact, he had already started work on his doctoral dissertation in Algebraic Topology.

  Convinced as I was until that point that the wounds of my short traumatic history as a mathematics hopeful had more or less healed, I was delighted, even amused, when I learned the identity of my new room-mate. As we were dining side by side in the university dining hall on our first evening, to get better acquainted, I said to him casually:

  'Since you're a mathematical genius, Sammy, I'm sure you can easily prove that every even number greater than 2 is th
e sum of two primes.'

  He burst out laughing. 'If I could prove that, man, I wouldn't be here eating with you; I'd be a professor already. Maybe I'd even have my Fields Medal, the Nobel Prize of Mathematics!'

  Even as he was speaking, in a flash of revelation, I guessed the awful truth. Sammy confirmed it with his next words:

  'The statement you just made is Goldbach's Conjecture, one of the most notoriously difficult unsolved problems in the whole of mathematics!'

  My reactions went through the phases referred to (if I accurately remember what I learned in my elementary college Psychology course) as the Four Stages of Mourning: Denial, Anger, Depression and Acceptance.

  Of these, the first was the most short-lived. 'It… it can't be!' I stammered as soon as Sammy had uttered the horrible words, hoping I'd misheard.

  'What do you mean "it can't be"?' he asked. 'It can and it is! Goldbach's Conjecture – that's the name of the hypothesis, for it is only a hypothesis, since it's never been proved – is that all evens are the sum of two primes. It was first stated by a mathematician named Goldbach in a letter to Euler [3]. Although it's been tested and found to be true up to enormous even numbers, no one has managed to find a general proof.'

  I didn't hear Sammy's next words, for I had already passed into the stage of Anger:

  'The old bastard!' I yelled in Greek. "The son of a bitch! God damn him! May he rot in hell!'

  My new room-mate, totally bewildered that a hypothesis in Number Theory could provoke such an outburst of violent Mediterranean passion, pleaded with me to tell him what was going on. I, however, was in no state for explanations.

  I was nineteen and until then had led a protected life. Except for the single Scotch drunk with Father to celebrate, 'among grown-up men', my graduation from high school and the required sip of wine to toast a relative's wedding, I had never tasted alcohol. Consequently, the great quantities I put down that night at a bar near the university (I started out with beer, moved on to bourbon and ended up with rum) must be multiplied by a rather large n to fully realize their effect.