THE LAST THEOREM ( uncorrected proof)

Book by Clarke Arthur C Pohl Frederik

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 o...