THE LAST THEOREM

Arthur C. Clarke & Frederik Pohl

Contents

The First Preamble The Second Preamble The Third Preamble: The Last Theorem Chapter One: On Swami Rock Chapter Two: University Chapter Three: An Adventure In Code-Cracking Chapter Four: Forty Days Of Data Downpour Chapter Five: From Mercury To The Oort Chapter Six: Meanwhile, Back On Earth Chapter Seven: Getting There Chapter Eight: Summer Chapter Nine: Lazy Days Chapter Ten: A New Life For The Kanakaratnams Chapter Eleven: Pirate Life Chapter Twelve: Judgment Chapter Thirteen: A Convenient Place For Questioning Chapter Fourteen: Rendition To The Highest Bidder Chapter Fifteen: Introduction To One (Or More) Grand Galactics Chapter Sixteen: Homegoing Chapter Seventeen: Heaven Chapter Eighteen: Company Chapter Nineteen: Fame Chapter Twenty: Marriage Chapter Twenty One: Honeymoon, Part Two Chapter Twenty Two: The New World Chapter Twenty Three: Farmer “Bill” Chapter Twenty Four: California Chapter Twenty Five: Silent Thunder Chapter Twenty Six: On The Threshold Of Peace Chapter Twenty Seven: Pax Per Fidem Chapter Twenty Eight: Making A Life Chapter Twenty Nine: Burgeoning Hopes Chapter Thirty: Big News Chapter Thirty One: Skyhook Days Chapter Thirty Two: Natasha’s Gold Chapter Thirty Three: Private Pain In A Rejoicing World Chapter Thirty Four: Pentominoes And Cars Chapter Thirty Five: The Uses Of Vaccination Chapter Thirty Six: Preparing For The Race Chapter Thirty Seven: The Race Chapter Thirty Eight: The Hunt For Natasha Subramanian Chapter Thirty Nine: The Interrogations Chapter Forty: The Portrait Gallery Chapter Forty One: Home Again Chapter Forty Two: A Great Depression Chapter Forty Three: Landed Immigrants Chapter Forty Four: International Disagreements Chapter Forty Five: Searching For A Solution Chapter Forty Six: Deal-Making Chapter Forty Seven: Parting Chapter Forty Eight: The Soul In The Machine The First Postamble The Second Postamble The Third Postamble The Fourth Postamble By Arthur C. Clarke By Frederik Pohl Copyright About the Publisher

THE FIRST PREAMBLE

Arthur C. Clarke says:

The incidents at Pearl Harbor lay in the future and the United States was still at peace when a British warship steamed into Nantucket with what was later called “the most valuable cargo ever to reach American shores.” It was not very impressive, a metal cylinder about an inch high, fitted with connections and cooling fins. It could easily be carried in one hand. Yet this small object had a strong claim to being responsible for winning the war in Europe and Asia—though it did take the atom bomb to finish the last of the Axis powers off.

That just-invented object was the cavity magnetron.

The magnetron was not in principle a new idea. For some time it had been known that a powerful magnetic field could keep electrons racing in tight circles, thus generating radio waves. However, this fact remained little more than a laboratory curiosity until it was realized that those radio waves could be used for a military purpose.

When it had such a military use, it was called radar.

When the American scientists working at the Massachusetts Institute of Technology received that first device, they subjected it to many tests. They were surprised to find that the magnetron’s power output was so great that none of their laboratory instruments could measure it. A little later, powering the giant antennae that had quickly been erected along the Channel coast, that British radar did a fine job of spotting the Luftwaffe’s myriad warplanes as they formed up to attack England. Indeed, radar was responsible, more than any other one thing, for allowing the Royal Air Force to win the Battle of Britain.

It was soon realized that radar could be used not only to detect enemy aircraft in the sky, but to make electronic maps of the ground over which a plane was flying. That meant that, even in total darkness or complete overcast, the land below could be imaged in recognizable shape on a cathode-ray tube, thus helping navigation—and bombing missions. And as soon as the magnetron was available at MIT, a team headed by future Nobel Laureate Luis Alvarez asked the next question: “Can’t we use radar to land aircraft safely, as well as to shoot them down?”

So began GCA, or ground-controlled approach, the landing of aircraft in bad weather using precision approach radar.

The experimental Mark 1 GCA used two separate radars, one working at ten centimeters to locate the plane’s direction in azimuth, and the other—the world’s first three-centimeter radar—to measure height above ground. An operator seated before the two screens could then talk the aircraft down, telling the pilot when to fly right or left—or sometimes, more urgently, when to increase altitude—fast.

GCA was welcomed enthusiastically by the RAF Bomber Command, which every day was losing more aircraft over Europe through bad weather than through enemy action. In 1943 the Mark 1 and its crew were stationed at an airfield in St. Eval, Cornwall. An RAF crew headed by Flight Lieutenant Lavington was dispatched to join them. Lavington was assisted by the newly commissioned Pilot Officer Arthur C. Clarke.

Actually, Clarke should not have been in the Royal Air Force at all. As a civilian he had been a civil servant in H.M. Exchequer and Audit Department and hence had been in a reserved occupation. However, he had rightly suspected that he would soon be unreserved, so one day he sneaked away from the office and volunteered at the nearest RAF recruiting station. He was just in time. A few weeks later the army started looking for him—as an army deserter who was wanted by the medical corps! As he was unable to bear the sight of blood, particularly his own, he obviously had a very narrow escape.

By that time Arthur Clarke was already a keen space-cadet, having joined the British Interplanetary Society soon after it was formed in 1933. Now, realizing that he had at his command the world’s most powerful radar, producing beams only a fraction of a degree wide, one night he aimed it at the rising moon and counted for three seconds to see if there would be a returning echo.

Sadly, there wasn’t. It was years later before anyone did actually receive radar echoes from the moon.

However, although no one could have known it at the time, something else might have happened.

THE SECOND PREAMBLE

Frederik Pohl says:

There are two things in my life that I think have a bearing on the subject matter of this book, so perhaps this would be a good time to set them down.

The first: By the time I was in my early thirties, I had been exposed to a fair amount of mathematics—algebra, geometry, trigonometry, a little elementary calculus—either at Brooklyn Tech, where for a brief period in my youth I had the mistaken notion that I might become a chemical engineer, or, during World War II, in the U.S. Air Force Weather School at Chanute Field in Illinois, where the instructors tried to teach me something about the mathematical bases of meteorology.

None of those kinds of math made a great impression on me. What changed that, radically and permanently, was an article in Scientific American in the early 1950s that spoke of a sort of mathematics I had never before heard of. It was called “number theory.” It had to do with describing and cataloging the properties of that basic unit of all mathematics, the number, and it tickled my imagination.

I sent my secretary out to the nearest bookstore to buy me copies of all the books cited in the article, and I read them, and I was addicted. Over the next year and more I spent all the time I could squeeze out of a busy life in scribbling interminable calculations on ream upon ream of paper. (We’re talking about the 1950s, remember. No computers. Not even a pocket calculator. If I wanted to try factoring a number that I thought might be prime, I did it the way Fermat or Kepler or, for that matter, probably old Aristarchus himself had done it, which is to say, by means of interminably repetitious and laborious handwritten arithmetic.)

I never did find Fermat’s lost proof, or solve any other of the great mathematical puzzles. I didn’t even get very far with the one endeavor that, I thought for a time, I might actually make some headway with, namely, finding a formula for generating prime numbers. What I did accomplish—and little enough it is, for all that work—was to invent a couple of what you might call mathematical parlor tricks. One was a technique for counting on your fingers. (Hey, anybody can count on his fingers, you say. Well, sure, but up to 1,023?) The other was accomplishing an apparently impossible task.

I’ll give you the patter that goes with that trick:

If you draw a row of coins, it doesn’t matter how long a row, I will in ten seconds or less write down the exact number of permutations (heads-tails-heads, heads-tails-tails, etc.) that number of coins produces when flipped. And just to make it a little tougher on me, I will do it even if you cover up as many coins in the row as you like, from either end, so that I won’t be able to tell how many there are in the row.

Impossible, right? Care to try to figure it out? I’ll come back to you on this, but not right away.

The second thing that I think might be relevant happened some twenty years later, when I found myself for the first time in my life spending a few weeks in the island empire of Japan. I was there as a guest of Japanese science-fiction fandom, and so was Brian Aldiss, representing Britain, Yuli Kagarlitski, representing what then was still the Union of Soviet Socialist Republics, Judith Merril, representing Canada, and Arthur C. Clarke, representing Sri Lanka and most of the rest of the inhabited parts of the earth. Along with a contingent of Japanese writers and editors, the bunch of us had been touring Japanese cities, lecturing, being interviewed, and, on request, showing our silly sides. (Arthur did a sort of Sri Lankan version of a Hawaiian hula. Brian got involved in trying to pronounce a long list of Japanese words, most of which—for our hosts loved a good prank—turned out to be violently obscene. I won’t tell you what I did.) For a reward we were all treated to a decompressing weekend on Lake Biwa, where we lounged about in our kimonos and depleted the hotel’s bar.

We spent most of the time catching one another up on what we’d been doing since the last time we had been together. I thought Judy Merril had the most interesting story to tell. She had come early to Japan, and had sneaked a couple of days in Hiroshima before the rest of us had arrived. She was good at describing things, too, and she kept us interested while she told us what she had seen. Well, everyone knows about the twisted ironwork the Japanese preserved as a memorial when every other part of that building had been blown away by that first-ever-deployed-in-anger nuclear bomb, and about the melted-down face on the stone Buddha. And everyone knows about—the one that nobody can forget once that picture enters their minds—the shadow of a man that had been permanently etched, onto the stone stairs where he had been sitting, by the intolerably brilliant nuclear blast from the overhead sky.

“That must have been bright,” someone said—I think Brian.

Arthur said, “Bright enough that it could have been seen by a dozen nearby stars by now.”

“If anyone lives there to be looking,” someone else said—I think it was me.

And, we agreed, maybe someone might indeed be looking… or at least it was pretty to think so.

As to those mathematical parlor tricks:

I don’t think I should explain them to you just yet, but I promise that before this book is ended, someone will.

That someone will probably be a bright young man named Ranjit Subramanian, whom you are bound to meet in just a few pages.

After all, this book is basically Ranjit’s story.

THE THIRD PREAMBLE

Atmospheric Testing

In the spring of the year 1946, in a (previously) unspoiled South Pacific atoll named Bikini, the American navy put together a fleet of ninety-odd vessels. They were battleships, cruisers, destroyers, submarines, and assorted support craft, and they came from many sources. Some were captured German or Japanese ships, the spoils of battle from the recently ended World War II; most were war-weary or technologically outmoded American vessels.

This fleet was not meant to sail off into a giant naval battle against anyone, or indeed to go anywhere at all. Bikini Atoll was the vessels’ last stop. The reason the fleet had been assembled was simply so that a couple of atomic bombs could be inflicted on it. One would come from the air, the other from under the sea. The hope was that this travail could give the admirals some idea of what their navy might suffer in some future nuclear war.

Bikini Atoll, of course, wasn’t the end of nuclear weapons testing. It was just the beginning. Over the next dozen years and more the Americans exploded bomb after bomb in the atmosphere, diligently noting yield and damage done and every other number that could be extracted from a test. So did the Soviets and the Brits a little later, the French and the Chinese later still. Altogether the first five nuclear powers (who also happened, not by chance, to be the five permanent members of the United Nations Security Council) set off a total of more than fifteen hundred nuclear weapons in the atmosphere. They did this in places such as the Marshall Islands in the Pacific, in Algeria and French Polynesia, in desert areas of Australia, in Semipalatinsk in Soviet Kazakhstan and Novaya Zemlya in the Arctic Ocean, in the marshy waste of Lop Nor in China, and in many other places all around the world.

It did not greatly matter where the blasts originated. Each one of them produced an inconceivably brilliant flash— “brighter than a thousand suns,” was how the physicist Hans Thirring described it—a flash that swelled out into space in a hemispheric shell of photons, expanding at the rate of three hundred thousand kilometers in each second.

By then the photons of that first puny radar flash that young Arthur Clarke had aimed at the moon had traveled a long way from the place in the galaxy where Earth had been when the photons were launched.

How far had they gone? Well, by then it had been some thirty years since that radar flash of his had returned no data. Light—or radio waves, or electronic radiation of any kind—travels at, well, at that velocity of 186,000 miles (or some 300,000 kilometers) per second that is the speed of light. So each year those photons had traveled one light-year farther away, and in their passage they had already swept through the systems of several hundred stars. Many of those stars had planets. A few had planets capable of supporting life. A small fraction of that life was intelligent.

Human beings never did know which star’s beings first detected what was happening on Earth. Was it Groombridge 1618? Alpha Centauri B? (Or, for that matter, A?) Or Lalande 21185, Epsilon Eridani, perhaps even Tau Ceti?

The humans never knew, and perhaps that was just as well. It would only have worried them.

Whichever star’s system they had inhabited, the astronomers among those creatures (they didn’t call themselves astronomers; their term for what they did was something like “inventoriers of externalities”) paid close attention to that first weak pulse. It troubled them.

These people did not look in the least like humans, but they had certain nearly human “emotions,” and among them was something quite like fear. The microwave emanations from Earth were the first things to worry them. Then came those far brighter fire bursts that arrived a little later—the ones from the first nuclear test from White Sands Proving Ground, and then from Hiroshima and Nagasaki, and then from everywhere. Those flashes set those alien skywatchers to clattering and squealing at one another. Such flashes represented trouble, and potentially very big trouble.

It wasn’t that those first observers felt fear at what humanity was doing on its remote little planet. They didn’t care in the least what happened to planet Earth. What worried them was that that same expanding hemisphere of radiation would not die out once it passed their star. It would keep going on, farther and deeper into the galaxy. And sooner or later it would reach certain other individuals who were likely to take it very seriously indeed.

THE LAST THEOREM

1

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. It was the end of the semester and, 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 adorned 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, who with no formal training in mathematics, 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 Fermat’s 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. 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?”

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. “But 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 people 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 two-thousand-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.

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