The Last Theorem

Chapter One


ON SWAMI ROCK

And so now, at last, we meet this Ranjit Subramanian, the one whose long and remarkable life this book is all about. 

At this time Ranjit was sixteen years old, a freshman at Sri Lanka’s principal university, in the city of Colombo, and more full of himself than even your average sixteen- year- old. He wasn’t at the university now, though. At his father’s bidding he had made the long trip from Colombo slantwise across the island of Sri Lanka to the district of Trincomalee, where his father had the distinction of being chief priest at the Hindu temple called Tiru Koneswaram. Ranjit actually loved his father very much. 

He was almost always glad to see him. This time, however, he was a bit less so, because this time Ranjit had a pretty good idea of what the revered Ganesh Subramanian wanted to talk to him about. 

Ranjit was an intelligent boy, in fact one who was quite close to being as smart as he thought he was. He was a good- looking one, too. He wasn’t terribly tall, but most Sri Lankans aren’t. Ethnically he was a Tamil, and his skin color was the rich dark brown of a spoonful of cocoa powder, just before it went into the hot milk. The skin color wasn’t because he was a Tamil, though. Sri Lankans have a rich palette of complexions from near- Scandinavian white to a black so dark it seems almost purple. Ranjit’s best friend, Gamini Bandara, was pure Sinhalese for as many generations back as anyone had bothered to count, but the boys were the same in skin hue. The boys had been friends for a long time—since that scary night when Gamini’s school had burned to the ground, probably put to the torch by a couple of upperclassmen smoking forbidden cigarettes in a storage room. 

Like every other nearby human being capable of picking up a splintered piece of plywood and throwing it on the back of a truck, Ranjit had been drafted for emergency relief work. So had all the rest of the student body of his own school. It had been a dirty job, a lot harder than a youngster’s developing muscles were used to, not to mention the splinters and the scrapes and the endless cuts from the broken glass that was everywhere. Those were the bad parts, and there were plenty of them. But there were good parts, too. Like the time when Ranjit and some other boy around his own age finally got down to the source of some plaintive sounds that were coming from a debris pile, and released the headmaster’s terrified, but intact, elderly Siamese cat. 

When a teacher had carried the cat off to its owner, the two boys had stood grinning at each other. Ranjit had stuck his hand out, English fashion. “I’m Ranjit Subramanian,” he’d said. 

“And I’m Gamini Bandara,” the other boy had said, pumping his hand gleefully, “and, hey, we did a pretty good job here, didn’t we?” 

They agreed that they had. When at last they had been allowed to quit work for the day, they had lined up together for the sort of porridge that was their evening meal, and plopped their sleeping bags next to each other that night, and they had been best friends ever since. Helped out, to be sure, by the fact that Gamini’s school had been made uninhabitable by the fire and so its students had to double up at Ranjit’s. Gamini turned out to be pretty much everything a best friend could be, including the fact that the one great obsession in Ranjit’s life, the one for which there was no room for another person to share, didn’t interest Gamini at all. 

And, of course, there was one other thing that Gamini was. That was the part of Ranjit’s impending talk with his father that Ranjit least wanted to have. 

Ranjit grimaced to himself. As instructed, Ranjit went straight to one of the temple’s side doors, but it wasn’t his father who met him there. It was an elderly monk named Surash who told Ranjit—rather officiously, Ranjit thought—only that he would have to wait a bit. So Ranjit waited, for what he considered quite a long time, with nothing to do but listen to the bustle that came from within his father’s temple, about which Ranjit had mixed emotions. 

The temple had given his father purpose, position, and a rewarding career, all of which was good. However, it had also encouraged the old man in the vain hope that his son would follow in his footsteps. That was not going to happen. Even as a boy, Ranjit had not been able to believe in the complex Hindu pantheon of gods and goddesses, some with their various animal heads and unusual number of arms, whose sculptured figures encrusted the temple walls. Ranjit had been able to name every one of them, and to list its special powers and principal fast days as well, by the time he was six. It hadn’t been out of religious fervor. It had been simply because he had wanted to please the father he loved. 

Ranjit remembered waking early in the morning when he was a small child, still living at home, and his father getting up at sunrise to bathe in the temple pool. He would see his father, naked to the waist as he faced the rising sun, and hear his long, reverberating Om. When he was a little older, Ranjit himself learned to say the mantra, and the location of the six parts of the body that he touched, and to offer water to the statues in the puja room. But then he went away to school. His religious observances were not required, and therefore ended. By the time he was ten, he knew he would never follow in his father’s faith. 

Not that his father’s was not a fine profession. True, Ganesh Subramanian’s temple was neither as ancient nor as vast as the one it had attempted to replace. Although it had been bravely given the same name as the original—Tiru Koneswaram—even its chief priest rarely called it anything but “the new temple.” It hadn’t been completed until 1983, and in size it was not a patch on the original Tiru Koneswaram, the famous “temple with a thousand columns,” whose beginnings had been shrouded by two thousand years of history. 

And then, when at last Ranjit was met, it was not by his father but by old Surash. He was apologetic. “It is these pilgrims,” he said. “So many of them! More than one hundred, and your father, the chief priest, is determined to greet each one. Go, Ranjit. Sit on Swami Rock and watch the sea. In an hour, perhaps, your father will join you there, but just now—” He sighed, and shook his head, and turned away to the task of helping his boss cope with the flood of pilgrims. Leaving Ranjit to his own resources. Which, as a matter of fact, was just fine, because for Ranjit an hour or so to himself on Swami Rock was a welcome gift. 

An hour or so earlier Swami Rock would have been crowded with couples and whole families picnicking, sightseeing, or simply enjoying the cooling breeze that came off the Bay of Bengal. Now, with the sun lowering behind the hills to the west, it was almost deserted. 

That was the way Ranjit preferred it. He loved Swami Rock. Had loved it all his life, in fact—or no, he amended the thought, at six or seven he hadn’t actually loved the rock itself nearly as much as he had the surrounding lagoons and beaches, where you could catch little star tortoises and make them race against one another. 

But that was then. Now, at sixteen, he considered himself a fully adult man, and he had more important things to think about. 

Ranjit found an unoccupied stone bench and leaned back, enjoying both the warmth of the setting sun at his back and the sea breeze on his face, as he prepared to think about the two subjects that were on his mind. The first, actually, took little thinking. Ranjit wasn’t really disappointed at his father’s absence. Ganesh had not told his sixteen- year- old son just what it was that he wanted to discuss. Ranjit, however, was depressingly confident that he knew what it was. 

What it was was an embarrassment, and the worst part of it was that it was a wholly unnecessary one. It could have been avoided entirely if he had only remembered to lock his bedroom door so that the porter at his university lodgings would not have been able to blunder in on the two of them that afternoon. But Ranjit hadn’t locked his door. The porter had indeed walked in on them, and Ranjit knew that Ganesh Subramanian had long since interviewed the man. He had talked to the porter only for the purpose, Ganesh would have said, of making sure that Ranjit lacked nothing he needed. But it did carry the collateral benefit of ensuring that Ganesh was kept well informed of what was going on in his son’s life. Ranjit sighed. He would have wished to avoid the coming discussion. But he couldn’t, and so he turned his attention to the second subject on his mind—the important one—the one that was nearly always at the top of his thoughts. 

From his perch atop Swami Rock, a hundred meters above the restless waters of the Bay of Bengal, he looked eastward. On the surface, at twilight, there was nothing to see but water—in fact nothing at all for more than a thousand kilometers, apart from a few scattered islands, until you reached the coast of Thailand. Tonight there had been a lull in the northeast monsoon, and the sky was perfectly clear. A brilliant star, its light slightly tinged with orangey- red, lay low in the east, the brightest star in the sky. Idly, Ranjit wondered what it was named. His father would know, of course. Ganesh Subramanian was a devout and sincere believer in astrology, as a temple priest should be. But he had also had a lifelong interest in secular science of all kinds. He knew the planets of the solar system, and the names of many of the elements, and how it was that a few rods of metallic uranium could be made to manufacture the electrical power that could light a city, and he had passed some of that love on to his son. What remained with Ranjit, though, was not so much about the astronomy and physics and biology of the world, but most of all that one subject that bound everything else together, mathematics. 

That, Ranjit knew, he owed to his father because of the book his father had given him on his thirteenth birthday. The book was G. H. Hardy’s A Mathematician’s Apology. It was in that book that Ranjit first encountered the name of Srinivasa Ramanujan, the impoverished Indian clerk, with no formal training in mathematics, who had been the wonder of the mathematical world in the dark years of World War I. It was Hardy who received a letter from Ramanujan with some hundred of the theorems he had discovered, and Hardy who brought him to England and to world fame. 

Ramanujan was an inspiration to Ranjit—clearly mathematical genius could come from anyone—and the book had left him with a specific, dominating interest in number theory. Not just number theory: in particular the wonderful insights that were the work of the centuries- old genius Pierre de Fermat, and even more in particular that towering question Fermat had left for his successors, the proof—or the proof that there was no proof—of Fermat’s celebrated Last Theorem. 

That was Ranjit’s obsession, and it was the subject he proposed to devote the next hour to thinking about. It was too bad that he didn’t have his calculator in his pocket, but his best friend had talked him out of that. “You remember my cousin Charitha?” Gamini had said. “The one who is a captain in the army? He says that some of the guards in the trains are confiscating calculators. They sell them for what they can get. Your two- hundred- dollar Texas Instruments calculator they would sell for perhaps ten dollars to somebody who only wants to keep track of his cash outlays, so leave it at home.” Which Ranjit sensibly had done. 

The calculator’s absence was an annoyance, but not a particularly important one, for the wonderful thing about Fermat’s Last Theorem was its simplicity. After all, what could be simpler than a2 + b2 = c2? That is, the length of one arm of a right triangle, squared, added to the squared length of the other arm equals the square of the hypotenuse. (The simplest case is when the arms are three units and four units in length and the hypotenuse is then five units, but there are many other cases with unitary answers.) This simple equation anyone could prove for himself with a ruler and a little arithmetic. What Fermat had done to obsess generations of mathematicians was to claim that such a relationship worked only for squares, not for cubes or for any higher power. He could prove it, he said. 

But he didn’t publish his proof. 

(If you would like a fuller discussion of the “last” theorem, one is included at the end of this book, under the title “The Third Postamble.”) Ranjit stretched, yawned, and shook himself out of his reverie. He picked up a pebble and threw it as hard as he could, losing sight of it in the dusk long before it struck the water below. He smiled. All right, he confessed to himself, some part of what he knew other people said about him wasn’t totally untrue. For instance, it wasn’t entirely wrong to say that he was obsessed. He had chosen his loyalties early, and he stayed with them, and now he was what one might call a Fermatian. If Fermat claimed he had a proof, then Ranjit Subramanian, like many a mathematician before him, took it as an article of faith that that proof did exist. 

By that, however, Ranjit certainly did not mean an aberration like the so- called Wiles proof that he had tried to get his math professor to discuss at the university. If that cumbersome old turkey (it dated from the closing years of the twentieth century) could be called a proof at all—and Ranjit hesitated to use “proof ” for something no biological human could read— Ranjit didn’t deny its technical validity. He simply thought it was trash. In fact, as he had told Gamini Bandara just before that confounded porter had opened the door on the two of them, it certainly was not the proof that Pierre de Fermat had boasted of when he’d scribbled in the margin of his volume of Diophantus’s Arithmetica. 

Ranjit grinned again, wryly, because the next thing he had said to Gamini was that he was going to find Fermat’s proof for himself. And that was what had started the laughing put- downs and the friendly horseplay that had led directly to what the porter had walked in on. And Ranjit’s mind was so filled with the memories of that time that he never heard his father’s footsteps, and didn’t know his father was there until the old man put a hand on his shoulder and said, “Lost in thought, is that what you are?” 

The pressure of Ganesh’s hand kept his son from rising. Ganesh seated himself beside him, methodically studying Ranjit’s face, dress, and body. “You are thin,” he complained. 

“So are you,” Ranjit told him, smiling, but a little worried, too, because on his father’s face was a look he had never seen before, a worry and a sorrow that did not befit the usually upbeat old man. He added, “Don’t worry. They feed me well enough at the university.” 

His father nodded. “Yes,” he said, acknowledging the accuracy of the statement as well as the fact that he knew quite well just how adequately his son was fed. “Tell me what else they do for you there.” 

That might have been taken to be an invitation to say something about a boy’s right to a personal life and some freedom from being spied upon by servants. Ranjit elected to postpone that subject as long as he could. “Mainly,” he said, improvising hastily, “it’s been math that has kept me busy. You know about Fermat’s Last Theorem—” And then, when the look on Ganesh’s face showed real amusement for the first time, Ranjit said, “Well, of course you do. You’re the one who gave me the Hardy book in the first place, aren’t you? Anyway, there’s this so- called proof of Wiles. It’s an abomination. How does Wiles construct his proof? He goes back to Ken Ribet’s announcement that he had proved a link between Fermat and Taniyama- Shimura. That’s a conjecture that says—” 

Ganesh patted his shoulder. “Yes, Ranjit,” he said gently. “You needn’t bother to try to explain this Taniyama- Shimura thing to me.” 

“All right.” Ranjit thought for a moment. “Well, I’ll make it simple. 

The crux of Wiles’s argument lies in two theorems. The first is that a particular elliptical curve is semi- stable but it isn’t modular. The second says that all semi- stable elliptical curves that possess rational coefficients really are modular. That means there is a flat contradiction, and—” Ganesh sighed fondly. “You are really deeply involved in this, aren’t you?” he observed. “But you know your mathematics is far beyond me, so let’s talk about something else. What about the rest of your studies?” “Ah,” Ranjit said, faintly puzzled; it was not to talk about his classes, he was quite sure, that his father had brought him to Trincomalee. “Yes. My other classes.” As conversational subjects went, that one was not nearly as bad as the one about what the porter would have passed along. It wasn’t really wonderful, though. Ranjit sighed and bit the bullet. “Really,” he said, “why must I learn French? So I can go to the airport and sell souvenirs to tourists from Madagascar or Quebec?” 

His father smiled. “French is a language of culture,” he pointed out. “And also of your hero, Monsieur Fermat.” 

“Huh,” Ranjit said, recognizing the debating point but still unconvinced. “All right, but what about history? Who cares? Why do I need to know what the king of Kandy said to the Portuguese? Or whether the Dutch threw the English out of Trinco, or the other way around?” 

His father patted him again. “There is an easy answer to your question. The reason is because the university requires those credits of you before they will grant you your degree. After that, in graduate school, you can specialize as much as you like. Isn’t the university teaching you anything you enjoy other than math?” 

Ranjit brightened slightly. “Not now, no, but by next year I’ll be through with this really boring biology. Then I can take a different science course, and I’m going to do astronomy.” Reminded, he glanced up at the bright red star, now dominating the eastern horizon. 

His father did not disappoint him. “Yes, that’s Mars,” he said, following Ranjit’s line of sight. “It’s unusually bright; there’s good seeing tonight.” He turned his gaze back to his son. “Speaking of the planet Mars, do you remember who Percy Molesworth was? The one whose grave we used to visit?” 

Ranjit reached back into his recollections of childhood and was pleased to find a clue. “Oh, right. The astronomer.” They were speaking of Percy Molesworth, the British army captain who had been stationed at Trincomalee around the end of the nineteenth century. “Mars was his specialty, right?” he went on, happy to be talking about something that would please his father. “He was the one who proved that, uh . . .” 

“The canals,” his father assisted. 

“Right, the canals! He proved that they weren’t actual canals built by an advanced Martian civilization but just an example of the kind of tricks our eyes can play on us.” 

Ganesh gave him an encouraging nod. “He was the astronomer—the very great astronomer—who did most of his work right here in Trinco, and he—” 

Then Ganesh stopped midsentence. He turned to peer into Ranjit’s face. Then he sighed. “Do you see what I am doing, Ranjit? I am delaying the inevitable. It was not to talk about astronomers that I asked you to come here tonight. What we must discuss is something a great deal more serious. That is your relationship with Gamini Bandara.” 

It had come. 

Ranjit took a deep breath before bursting out: “Father, believe me! It is not what you think! We just play at that sort of thing, Gamini and I. It means nothing.” 

Unexpectedly, his father looked surprised. “Means nothing? Of course what you were doing means nothing. Did you think I did not know all the ways in which young boys like to experiment with kinds of behavior?” He shook his head reproachfully, and then said in a burst, “You must believe me in this, Ranjit. It isn’t the experimenting with sexual behavior that matters. It is the person you were sharing it with.” His voice was stressed again, as though it were hard for these words to come out. “Remember, my son, you are a Tamil. Bandara is Sinhalese.” 

Ranjit’s first reaction was that he could not believe what he was hearing from his father’s lips. How could his father, who had always taught him that all men were brothers, say such things now? Ganesh Subramanian had been faithful to his beliefs in spite of the fact that the ethnic riots that began in the 1980s had left scars that would take generations to heal. Ganesh had lost close relatives to rampaging mobs. He himself had narrowly escaped death more than once. 

But that was ancient history. Ranjit hadn’t been born yet in those days—even his deceased mother had hardly been born yet—and for years now there had been a well- kept truce. Ranjit raised a hand. “Father,” he begged, “please! This is not like you. Gamini hasn’t murdered anyone.” Inexorably Ganesh Subramanian repeated the terrible words. “Gamini is Sinhalese.” 

“But Father! What about all the things you taught me? About that poem you made me learn by heart, the one from the Purananuru. ‘To us all towns are one, all men our kin, thus we have seen in the visions of the wise.’ ” 

He was clutching at straws. His father was not to be moved by twothousand- year- old Tamil verses. He didn’t answer, just shook his head, though Ranjit could see from the expression on his face that he was suffering, too. 

“All right,” Ranjit said miserably. “What do you want me to do?” 

His father’s voice was heavy. “What you must, Ranjit. You cannot remain so close to a Sinhalese.” 

“But why? Why now?” 

“I have no choice in this,” his father said. “I must put my duties as high priest of the temple first, and this matter is causing dissension.” He sighed and then said, “You were raised to be loyal, Ranjit. I am not surprised that you want to stand with your friend. I only hoped that you could find a way to be loyal to your father as well, but perhaps that is impossible.” He shook his head and then stood up, looking down at his son. “Ranjit,” he said, “I must tell you that you are not now welcome in my house. One of the monks will find you a place to sleep tonight. If you finally choose to sever your relationship with Bandara, phone or write me to tell me so. Until you do, there is no reason for you to contact me again.” 

As his father turned and walked away, Ranjit dropped quite suddenly into a state of misery. . . . 

Perhaps that state needs to be examined more closely. Ranjit was certainly miserable with the sudden distance that had opened between himself and his beloved father. Nothing in that fact, however, led him to think that he himself was in any way in the wrong. He was, after all, just sixteen years old. 

And some twenty light- years away, on a planet so corrupted and befouled that it was very difficult to believe any organic creature could survive there, an odd- looking race known as the One Point Fives was nevertheless surviving. 

The question now on the collective minds of the One Point Fives, as they prepared themselves to meet the inevitable orders from the Grand Galactics who were their masters, was how much longer that survival would go on. 

True, the One Point Fives hadn’t received their marching orders yet. But they knew what was coming. They themselves had detected the troublesome emissions from Earth as the successive waves of photons had swept by. They knew as well just when those photons would reach the Grand Galactics. 

Most of all, they knew just how the Grand Galactics were likely to respond. The thought of what that response might mean for them made them shudder within their body armor. 

The One Point Fives had only one real hope. That was that they would be able to accomplish everything the Grand Galactics would demand of them and, when that task was completed, that they would have enough survivors among their own people to keep the race alive.