Apostolos Doxiadis - Uncle Petros and Goldbach

'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. Leave 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.