The Last Theorem

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.