‘But Fermat’s last theorem is something different. It is, I believe, much easier to explain than those objects you have just mentioned. You are a teacher of children, Miss Duncan, so perhaps you have already encountered this simple question: can you think of three ordinary numbers, not zero, of course, such that the square of the first plus the square of the second is equal to the square of the third?’
‘Well, certainly; that’s Pythagoras’ theorem on right triangles,’ I answered; ‘three squared plus four squared equals five squared, for example.’
‘Very good, very good! Nine and sixteen make twenty-five,’ he nodded approvingly. ‘And there are many, many more; an infinite number, in fact. If you seek you will easily find a great many more. Thirty-six plus sixty-four equals one hundred – did you ever notice that? A delightful equation – such beautiful numbers. But now – now for the great mystery of Fermat. Can you think of three numbers such that the cube of the first plus the cube of the second is equal to the cube of the third?’
‘Let me see,’ I hesitated. ‘One cubed plus two cubed is nine – no, that’s a square. Two cubed plus …’
‘Do not try, do not try any longer, for it is impossible! No such numbers exist, none at all, and this was proved long ago, some say by Fermat himself, although his argument appears to contain an error, which was rectified over a century ago by the great Leonhard Euler. Fermat also left a proof for the fourth powers, a beautiful and astute argument. But he asserted that much more was true. Indeed, his famous so-called “theorem” states that one cannot find three numbers such that the nth power of the first plus the nth powers of the second is equal to the nth power of the third, for any ordinary number n greater than 2. You see, this is what you cannot have,’ and he scribbled:
xn + yn = zn
onto a bit of paper and showed it to me.
‘Really? You can never have that for any ordinary numbers x, y and z?’ I said, surprised, for the equation has nothing so very startling and improbable in its appearance.
‘That is the mystery! For Fermat himself wrote that he had proved that you cannot, but his proof was never found. He noted the formula on a page of his copy of Diophantus, saying that he had found a most marvellous proof, but that the margin was too narrow to contain it – cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.’
‘So you think he wrote it down elsewhere? And it has really never been found or rediscovered?’
‘Never, although some progress was made towards rediscovery at least. The greatest step of all was taken by a member of your sex, the great, great Sophie Germain, my inspirer and my muse in all things. Ah, the sublime beauty, the unspeakable wisdom of her method! She worked with prime numbers, certain very special prime numbers, the Germain primes. You know that a prime number, p, is a number which is not divisible by any other, save the number one, and itself, of course. Five, seven, eleven, these are primes. But the Germain primes are very special, for not only is p itself prime, but 2p + 1. Five is a prime number and so is twice five plus one – eleven, so five is a Germain prime! Seven is prime, but twice seven plus one is fifteen, not prime. So seven is not a Germain prime. The notion is so beautiful, so mysterious, so admirable. Inspired by this, I try and attempt to go even farther than she did!’
‘And what did she do, exactly? What did she use these Germain primes for?’
‘Very nearly, she proved that Fermat’s equation is impossible when n is one of them – you cannot have xp + yp = zp for a Germain prime p. Well, I exaggerate. She proved this in the case that x, y and z are not divisible by p. But no one has come close to proving anything so important in the subject, excluding the great Kummer of course, but that is something else, for his techniques are entirely modern and different, whereas hers could have been those of Fermat himself. Such purity, such simplicity! And she sent her beautiful theorem to the greatest mathematician of the day, Carl Friedrich Gauss, but she wrote under the identity of a man, for she feared that her sex might cause her discovery to be despised and rejected because of the prejudices of her time.’
‘Yes,’ I cried happily, recalling the story, ‘Monsieur Le Blanc! Charles told us how she assumed his identity in order to attend the École Polytechnique in Paris.’
‘Yes, and she used the same name to write to Gauss. But Gauss found out the truth, and contrary to what she had feared, he was filled with admiration. “A taste for the abstract sciences in general”,’ quoted Mr Korneck in a tone filled with reverence, ‘“and above all for the mysteries of numbers, is excessively rare: one is not astonished at it: the enchanting charms of this sublime science are revealed only to those who have the courage to study them deeply.” That is what he wrote to her; the sentence has been like a luminous guide to me. And he went on to tell her that if a woman was able to surmount all the difficulties and obstacles that customs and prejudices put in her way to study these mysteries, then, indeed, she must be a woman of superior genius. Such a man! Such understanding! If I could only write to him, but it is too late, for he died nearly forty years ago. I try to prove something similar to Germain, but even simpler and even more general, something that would work in all cases, something so surprising and beautiful that it may even be the secret proof of Fermat himself! I am working on it, working day and night. It does not quite succeed yet, but I feel that I am nearing the goal.’
‘Then you will be famous indeed,’ smiled Charles. ‘For great minds have worked hard over this problem and given up.’
‘Yes, yes, it has been so. But perhaps they tried too hard. Euler, Kummer – geniuses! They poured such a wealth of techniques and ideas into the problem as could not possibly have existed in Fermat’s mind. Something different is needed, something simpler; an illumination of perfection and simplicity. Ah, I dream of finding it – and I hope to live my dream! I am no distinguished professor of mathematics such as your honoured friends, Miss Duncan. I am a mere amateur; a lover of mathematics. I have not attended university nor written an erudite thesis. I come from beautiful Posen and grew up helping my father in the running of our family’s great horse farm, the raising of the most beautiful and powerful of Prussian steeds. These horses of mine come from our stables,’ he added with proud tenderness, shaking the reins over the backs of the white horses whose fresh trot was pulling us along smartly and effortlessly, and who answered his caress by tossing their heads and shaking their manes without breaking their rhythm in the slightest. ‘But I came upon the love of mathematics by myself,’ he continued, ‘although my passion was not so pleasing to my father, and have read the works of Fermat and Pascal and their followers in later generations, and now that I have reached an age at which the stables belong to me and are run by trustworthy men more able than I, and I find myself free to follow my dreams and inclinations, I devote myself to study. I spent many of the recent years in Paris, attending courses of the great mathematicians there, but decided some months ago to spend a year in Cambridge, for it is here that the greatest exponents of the subject of algebra are to be found.’
‘I hope you succeed,’ I cried enthusiastically. Poor Mr Korneck, he seemed very happy, but I detected a trace of bitterness in his words, as though he felt, in spite of all he had said, that his origins made him into a perpetual outsider in the world of mathematics that he loved. I forgave Charles all of his misdeeds in view of his kindness to the eager amateur, and resolved to restrain myself from upbraiding him more severely in private, as I had meant to do, with the intention of compelling him to greater discretion in the future.
Oh, how pleased I was when the carriage pulled into Cambridge, and we went trotting along the familiar streets! Oh, the delight when we started along the Chesterton Road, and Mrs Fitzpatrick’s house came into view! And how my heart beat as the door flew open and Arthur dashed out with uncharacteristic speed to welcome us home! Charles and Mr Korneck left us till the morrow, and I spent a blissful evening pouring out my heart in a great mixture of facts, thoughts, feelings and e
motions into Arthur’s most receptive ears. Together we analysed, discussed and reflected until there seemed quite nothing left to say – or perhaps, until silence seemed better to express what we wished to say than any words could do.
‘I have a surprise for you tomorrow,’ he told me finally, as at a late hour, he bid me goodnight. ‘It may be useful, and will certainly be interesting. No, I shan’t tell you more!’ And with twinkling eyes, he turned down the corridor and climbed up the stairs to his rooms.
I was tickled, and my first thought upon awakening this morning was that I should soon find out what Arthur had in store for me. He remained secretive throughout the morning, but at the approach of midday, he took my arm, and together we walked into the centre of town. Stopping in front of a pleasant restaurant, he looked inside, and then said, turning to me,
‘Ah, here is the person I wanted you to meet. He wants to meet you even more, so he’s bright and early!’
In we went, and sat down at a table already occupied by a very red-haired, freckled personage. I wondered greatly who he could be, and why it should be such a surprise for me to thus make his acquaintance, but I beamed upon him cheerfully, out of general feelings of bliss and human kindness that welled up in me purely from the pleasure of being with Arthur.
‘I am delighted, delighted to meet you, Miss Duncan!’ began the freckled young man, with a very pretty and musical (and perhaps even slightly exaggerated) Irish brogue. ‘Will you do the honours, Weatherburn, or must I introduce myself?’
‘Vanessa, this is Mr Patrick O’Sullivan,’ smiled Arthur, and he watched me carefully to see my reaction. It was a blank one for only the merest second.
‘Why, of course,’ I cried. ‘The journalist! I read your article on the murder of Mr Granger.’
‘Ah, that’s nothing, if you only knew!’ he said proudly. ‘How many articles I could write if I only could. I hear you’re involved in the case, Miss Duncan.’
‘Yes, I am interested,’ I answered, glancing at Arthur.
‘No need for euphemisms,’ he said seriously. ‘Vanessa has been hired to privately investigate the murder, as you well know. We needn’t mince words; Vanessa needs information, and you are certainly in possession of some of that indispensable commodity. I believe you should work together, for the sooner the whole story reaches its conclusion, the better.’
‘I need information too, yes I do indeed, it’s all part of my job. I’m looking for a scoop for my paper and for my reputation. I’m ready to consider a fair exchange, Miss Duncan: my information for yours.’
‘Oh – but I can’t really do that – I can’t give things I’m not even certain of, or which may not even have anything to do with the murder, to be published in a newspaper! Why, that is exactly why poor Mrs Bryce-Fortescue wants me to investigate – just exactly so that nothing may get into the papers!’
‘Well, that’s a forlorn hope, it’s ridiculous, it is, how many murders do we get about here that we should restrain ourselves from offering them up to our readers as a little delicacy?’
Arthur winced, and I glared at the tactless individual, but little wrinkles of laughter appeared at the corners of his eyes, so that it was difficult to feel seriously angry. I had opened my mouth to remonstrate, but closed it again. Was he so wrong? Did one not read such things in the papers with no more than a mild feeling of faint shock mingled with curiosity, when they did not concern oneself? I felt ashamed, remembering from my single past experience of such things, how much fear and bitter suffering was hidden between the lines of the articles the readers chatted about with casual interest before lightly flipping over the page.
‘Come now,’ said Mr O’Sullivan encouragingly. ‘Are you sure that your Mrs Bryce-Fortescue really wants to keep everything out of the papers altogether? Isn’t she just trying to protect her own family pride? It’s a funny thing, it is – some people’d kill to be in the papers—’ I glared ‘— oh, sorry – well, some people’d do just about anything to be in the papers, then, and others’d do as much and more to stay out of them. Odd things, newspapers, how they get the blood up. Love ’em, myself. Couldn’t live without ’em. Listen, then, Miss Duncan: here’s a bargain. I’ll tell you what I know, and you’ll understand exactly what’s threatening your Mrs Granger. You tell me what you know, and I’ll see that if Mrs Granger really is innocent, she stays out of the papers altogether. But you must understand that if she’s guilty, or even if she’s just arrested, I must put it in.’
‘How can you know anything about what is threatening her?’ I gasped, hearing only these words among the flood of his remarks.
‘Well, the long and short of it is that I’ve got a brother-in-law in the police force, in homicide, as a matter of fact. There, I’ve told you. I can find out just about anything the police know, for a promise not to spill any of it till they give me the wink that the time is ripe. But they don’t know much yet. That’s why I’m hoping you can tell me more. You see, this is how it is. If the police reach a solution, I won’t have a scoop, for when they’re sure, they’ll talk to journalists from all the papers. But if we get there before them, then we win! And if we use a little of their knowledge to help us get there, well, that’s all in the game.’
‘Oh!’ I said. ‘Dear me, as far as I personally am concerned, I don’t mind at all if they do solve it, as long as they do a proper job. But I’m not sure they are, for it seems to me that they really suspect Sylvia. You must know, then; is it true? They keep badgering her with threatening questions. Why, they even have a witness who claims that he saw her in the woods near the house just a few minutes before the murder took place, and she herself says it’s impossible, as she never left her room even for a moment. I for one believe her, but I think the police don’t! Am I right?’
‘It’s not easy to give a yes or no answer to the question. They suspect her, for certain. They don’t have a high opinion of her feeble alibi. There are too many ways to slip out of a house without using the main door. But they also believe she may possibly not have done the shooting herself, but have planned it, or know who did it. Their minds are made up that she either did it or knows who did, and they’re determined to get her, by accusing and arresting her without full proof, if necessary. As for their witness who saw her, it’s all true enough, except that the identification of Mrs Granger turned out to be pretty tentative. You know what I mean: “As I was passing the Granger estate, I saw Mrs Granger running through the trees.” But when the police insist, it becomes: “Well, I certainly saw someone that I thought was Mrs Granger.” So they ask her how much she could see, how far the person was, whether it was a man or a woman, how she could tell? And it turns out that the person was barely glimpsed between the leaves and branches. Well, the police believe someone was seen, but they were not so convinced by the identification of Mrs Granger. They just made out that they were to put the fear of God into her. Tricky, aren’t they?’
‘Tricky! That ought to be illegal!’ I cried indignantly, feeling a violent, irrational desire to protect Sylvia sweep over me. ‘Why, they’re frightening her half to death! Who was it that said she saw her, anyway? Do you know?’
‘Oh, it was some old cat that had her knife into your Sylvia. Spent more time massacring her character to the police and complaining about some occasion when she wouldn’t let her into the house, than stating what she actually saw. Her statement isn’t very helpful when you peel away everything but the facts. She hardly saw a thing.’
‘I know that lady – it’s Mrs Munn!’ I exclaimed. ‘And I wouldn’t take her word against Sylvia’s for a moment. The police may not either, but they certainly talked to Sylvia quite as though they did. It’s a shame! Well, they’re barking up the wrong tree. Sylvia is innocent.’
‘They seem pretty certain she had something to do with it,’ he said, looking at me closely.
‘They’re mistaken. Anybody can be wrongly accused and wrongly condemned,’ I said furiously. ‘How extraordinarily apt people are to forget t
hat!’
‘Yes, yes, of course. It can happen. We know that, we know that,’ he said hastily, glancing at Arthur. ‘All right, so the police can be crude in their ways. I’ll tell you the truth, since you ask. The police have it in for Mrs Sylvia Granger, whether or not she actually pulled the trigger. They are certain that she holds the key to the mystery, for the simple, telling reason that she hated her husband – and that she inherits everything he had! That’s how the police work: motive above all, and when you’ve got a love-hate motive and a financial motive all rolled up together, they won’t look at anything else. So either she did it, or she had an accomplice of some kind, some lover of hers. Clytemnestra, or whatever. And they’re ready to put the pressure on her, frighten her, accuse her, even put her in the dock, to make her talk. It’s hard lines, but it’s neither here nor there, right now. What we want – you and I – is to find out the truth, isn’t it? All right, Miss Duncan, I’ve told you what’s happening on the police side, and I see well enough that you don’t agree with them. You think she’s innocent, don’t you? I’ve played my cards. Aren’t you going to play some of yours? What do you know? What makes you believe what you believe?’
He turned a most winning smile upon me, but I resisted its charm, and compelled myself to think clearly about the bargain he proposed. On the favourable side, he clearly knew things of essential importance, and with time, as the police investigation progressed, he would know more. On the assumption that Sylvia was innocent, as I believed, anything we discovered would be likely to help her, and if she turned out to be guilty (even as an accomplice), then my failure could hardly matter. However, I feared his bargain because of his very quality as a journalist. I found it difficult to efface from my mind a distressing vision of Mrs Bryce-Fortescue furiously brandishing vulgar articles filled with the Bryce-Fortescue name in my face, and asking angrily who had provided the newspaper with the intimate details they contained!
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