The Last Theorem
However, not everything that had transpired had been bad. Under the close supervision of four of the five Scandinavian nations—Iceland, with its own domestic unrest, stayed outside the group—several of the most bitterly fought wars were in, however brief, remission. Even Myanmar, the country that was more commonly called Burma (except by its own intransigent governing clique), had without warning released all of its political prisoners and invited foreign diplomats to monitor its next set of elections. Finally—a development that would have greatly pleased Ranjit, if he could have known of it—after endless stalling, the World Bank had come through with a decent billion-dollar start-up grant for an actual Artsutanov space elevator. True, a World Bank grant was a long way from the actual wheels turning, with the cars going up and down the cables, the hardware that you could hop onto and be drawn to low earth orbit at three hundred kilometers an hour. But it was a real first step.
Those, of course, were not the only data with a significant bearing on his own life that Ranjit did not know. For example, he didn’t know why he had been taken to this place and why he had been tortured in it. And then, when the torturing had stopped, he didn’t know why that had happened, either. Ranjit had never heard of extraordinary rendition or the momentous decision that had been handed down, decades earlier, by the British Law Lords.
Of course, Ranjit’s torturers could have helped him out with some information if they had chosen to. They didn’t choose to.
After the first day without inflicted pain, he didn’t see Bruno, the belly-slap and electrical-cable guy, again at all. He did see Squinty quite often, but only after Squinty had extracted a promise from him that he would stop asking why he had been tortured and whether he would ever be released, and indeed pretty much any question that Ranjit really wanted answered. Squinty did supply a tiny bit of information. (“Bruno? Oh, he’s been promoted upstairs. He just doesn’t know what to do with a prisoner unless he’s hurting him, and it doesn’t look like we’re going to be hurting you anymore.”)
That was, Ranjit reminded himself, a fact of life not to be scorned. It was a big improvement over the previous diet of thrashings and water-boardings. But it was, especially after Squinty quit coming around because Ranjit couldn’t keep his promise to stop asking forbidden questions, pretty damn boring. Ranjit wasn’t left entirely without human company. There was a limping old man who brought him his food and carried away his slop buckets, but that one was no use for conversation. He no doubt spoke some language or other, but it didn’t seem to be one that Ranjit possessed.
Ranjit didn’t know when he first began to have long one-sided talks with his friends. With his absent friends, that is, since none of them was physically present in his cell.
Of course, none of them could hear what he said to them. It would have been interesting if, for example, Myra de Soyza could have, or Pru No-Name. Less interesting for Gamini Bandara because, after reporting on his own emptily monotonous existence, about all Ranjit had to say to his absent lifelong friend was that he really should have budgeted more time to be with Ranjit and less for the American woman, who, after all, would never see him again.
Some of Ranjit’s best absent friends were people he had never known in the flesh. For instance, there was the no-longer-living Paul Wolfskehl. Wolfskehl had been a nineteenth-century German business tycoon whose best-beloved sweetheart had turned down his proposal of marriage. That meant that, in spite of all his wealth and power, life was no longer worth living for Wolfskehl, so he sensibly decided to commit suicide. That didn’t work out, though. While Wolfskehl was waiting for the exact right moment to do himself in, he idly picked up a book to read.
The book chanced to concern Fermat’s Last Theorem, written by a man named Ernst Kummer. As it happened, Wolfskehl had attended a couple of Kummer’s lectures on number theory; curiosity made him read the new essay….
And, like many other amateur mathematicians, before and after, Wolfskehl was immediately hooked. He forgot about killing himself, being too busy trying to plumb the mysteries of a-squared plus b-squared equals c-squared, and the paradox that if the quantities were cubed, they never did equal each other.
Then there was the also long deceased Sophie Germain, whose teenage years had been spent during the frightening time of the French Revolution. Why this should have persuaded young Sophie to resolve on a career in mathematics is not immediately obvious. But it did.
Of course, that was not an easy ambition for a female to accomplish. As Elizabeth I of England had once put it, Sophie was cursed by being split rather than fringed, and so everything she tried to do was vastly harder for her than for her fringed colleagues.
Then, when his imaginary conversation partners ran out of steam, something Myra de Soyza had said began to cudgel Ranjit’s mind.
What had it been? Something about seeing what tools other mathematicians had possessed at the time Fermat had jotted his cursed boast in the margin of his book?
Well, what tools were they?
He remembered that Sophie Germain was said to have been the first mathematician of any gender to make any headway at all with the Fermat proof. So just what headway had she made?
Ranjit, of course, had no way of looking that up. Back at the university, equipped with a password, all he would have had to do was hit a few keys on the handiest computer and the damn woman’s entire life production would have been laid out for him to study.
But he didn’t have the computer. All he had was his memory, and he was not sure that it was adequate to the task at hand.
He did remember what a “Sophie Germain prime” was—that is, any prime, p, such that 2p + 1 was also a prime. Three was the littlest Sophie Germain prime: 3 × 2 + 1 = 7, and seven was a prime, all right. (Most of the other Sophie Germain primes were much larger, and thus hardly any fun at all.) Ranjit was quite pleased with himself for remembering this, though no matter how much he thought about it, he could not see any way in which a Sophie Germain prime could lead him to a solution of the Fermat problem.
There was one other thing. After profound labor Germain had produced a theorem of her own:
If x, y, and z are integers, and if x5 + y5 = z5, then x, y, or z must be divisible by five.
Like every other stepping-stone toward a proof that Ranjit had managed to quarry out of the refractory stone of his mind, this one was a disappointment. The equation made no sense. Fermat’s whole theorem was supposed to prove that no such equality as x5 + y5 = z5 could ever exist in the first place. So it wasn’t of any use at all….
Or was it? That is, forget Sophie Germain’s useless theorem itself, but how did she get to it?
And wasn’t that what Myra had suggested to him at Dr. Vorhulst’s party, back in the days when Ranjit could sometimes go to a party?
There was one other person (well, sort of person) with whom Ranjit had never, or never yet, had any personal dealings, but who (or which) could have supplied him with very useful data. It is probably about time that we spent a little more time with him (or them, or it, or maybe even her).
15
INTRODUCTION TO ONE (OR MORE) GRAND GALACTICS
The first thing we need to straighten out about this Grand Galactic person is whether in fact he was a he, or indeed a person, or, ultimately, “a” (rather than some fraction of “a”) Grand Galactic.
None of those questions has an easy answer. So what we’re going to do here is, we’re going to ignore the facts and settle for answers that are no problem for us to deal with, apart from their being just plain wrong. First, we will say that this person is really a person, in spite of also being a part of that larger “person” that was all the Grand Galactics combined.
There were Grand Galactics all over the place, from the galaxy’s accelerating fringes to its relatively motionless core, and just about everywhere in between. How many Grand Galactics were there? That’s also a meaningless question. There were many, many of them, but when you came right down to it, the many were also one, becaus
e whenever he chose, every Grand Galactic was instantly merged with any or every other.
As you have noticed, we just arbitrarily assigned gender to the Grand Galactics’ pronoun. Don’t assume from this that they practiced sexual intercourse in any sense understandable by a human being. They didn’t. It is just that we can’t go on with that “it” or “he” or “they” business forever, so we just cut the Gordian knot and made him a “he.”
We have just taken one rather large liberty. Let’s take another. Let’s give “him” a name. We’ll call him “Bill.” (Not Bill. “Bill.” It is a major liberty that we have taken, and we should acknowledge that we know it to be so by the use of the quotation marks.)
Now, what else would it be useful for us to know about the Grand Galactics at this time?
For example, would it be helpful to know how big they are? Or at least, since one node of Grand Galactics may be thousands, or billions, of light-years from some particular other node, how they measure distance?
Let’s assume it would indeed be helpful, but we must also recognize that, as with all questions about the Grand Galactics, the answer is going to be complicated. Start with the fact that the Grand Galactics don’t like the kind of arbitrary units of measurement human beings use. When you track those down, they are always based on some human value, such as the distance from a man’s fingertip to his armpit, or some calculated fraction of the distance from a pole to the equator on the particular planet that humans chance to occupy. Grand Galactic measurements are always made on the Planck scale, which is actually quite tiny. The measure of a single Planck unit is 1.616 × 10-35 meters. The easiest way to understand how little that is is to remember that it’s so small it is impossible to measure anything smaller.
(Why impossible? Because you can’t measure anything you can’t see, and nothing can be seen without employing those light-carrying particles called photons. And any photon that was powerful enough to illuminate a Planck-scale distance would be so extremely powerful—and thus so extremely massive—that it would immediately transform itself into a black hole. The word “impossible” is sometimes taken as a challenge. In this case, though, it’s just a fact.)
So to measure anything in any of the three dimensions, whether it’s the circumference of an electron or the diameter of the universe itself, Grand Galactics simply count the number of Planck distances along a line from point A to point B.
That is invariably a large number, but that’s all right with the Grand Galactics. Looked at in one way, they are pretty large numbers themselves.
So, having found ways of at least identifying the un-understandable, let’s get back to that much simpler being, Ranjit Subramanian.
When Ranjit was quite young, his highly ecumenical father encouraged him to read some rather strange books, one of which, by a writer named James Branch Cabell, was about the nature of writing and writers. (For a time Ganesh Subramanian thought that might be a career choice for his son.) What some would-be writers were trying to say to the world, Cabell wrote, was, “I am pregnant with words, and I must have lexicological parturition or I die.”
And, curiously, that is almost exactly the condition Ranjit now felt himself to be in.
For days now Ranjit had been pleading for help, shouting into the empty hallways, explaining, though no one seemed to be listening, that he had something that absolutely had to be communicated to a journal at once. There were no answers. Even the limping old man was now just putting Ranjit’s meals inside the door and limping away as fast as he could.
So when Ranjit heard the old man’s step-slide coming along the empty corridors, he felt little interest, except that this time there was, along with it, the rap-tap-tap of the footsteps of someone who wasn’t limping at all. A moment later Ranjit’s cell door opened. The old man was there, but deferentially a step or two behind another man—a man who wore an expression of shock and dismay on a face whose lineaments Ranjit knew as well as he knew his own. “Sweet God Almighty, Ranj,” Gamini Bandara said wonderingly, “is that really you?”
Of all the questions Ranjit might have asked this unexpected visitor from his past, he chose the simplest. “What are you doing here, Gamini?”
“What the hell do you think I’m doing? I’m going to get you out of here, and if you think that was easy, you’re crazier than you look. Then we’ll get you to a dentist—what happened to your front teeth? Or, no, I suppose first you need to see a doctor—What?”
Ranjit was standing now, almost quivering with excitement. “Not a doctor! If you can get me out of here, get me to a computer!”
Gamini looked puzzled. “A computer? Well, sure, that can be arranged, but first we need to make sure you’re all right—”
“Damn it, Gamini!” Ranjit cried. “Can’t you understand what I’m saying? I think I’ve got the proof! I need a computer, and I need it right now! Do you have any idea how terrified I am that I’ll forget some part of the proof before I can get it refereed?”
Ranjit got the doctor. He got the computer, too—in fact got both of them at the same time, but not until Gamini had walked him out of his prison to where a helicopter waited, its vane turning over. As Ranjit climbed into the chopper, he saw a couple of men standing nearby. Squinty was one of them; Squinty looked astonished and worried but didn’t even gesture a good-bye. Then a twenty-minute downhill flight, among great mountains that wore brilliant caps of ice. In the helicopter Ranjit could not help turning to Gamini with questions, but this time it was Gamini who didn’t want to talk. “Later,” he said, nodding at the chopper pilot, who wore a uniform Ranjit had never seen before.
They landed at a real airport, a scant dozen meters from a plane—and not just any plane, Ranjit saw, but a BAB-2200, the fastest and, in some configurations, the most luxurious aircraft Boeing-Airbus had ever built, and it wore the blue globe-and-wreath United Nations insignia. Inside, it was even more so. Its seating was in leather armchairs. And its crew consisted of a pilot (wearing the uniform of a colonel in the American air force) and two very pretty flight attendants (wearing the same uniforms but with captain’s bars, and over the uniform fluffy white aprons). “Heading for home now, sir?” the pilot asked Gamini. He got a nod for an answer and immediately disappeared into the cockpit. One of the attendants led Ranjit to a chair (which, he discovered, swiveled) and belted him in. “That’s Jeannie,” Gamini informed him, while himself being belted into another chair. “She’s a doctor, too, so you better let her check you out—”
“The computer—” Ranjit started to object.
“Oh, you’ll get your damn computer, Ranj, but first we have to get airborne. It’ll just be a minute.”
By then the two women had already retreated to their fold-down seats against a bulkhead and, sure enough, the plane had begun to move. And as soon as the seat belt sign went off, the second of the attendants—“I’m Amy. Hi!”—was magicking a laptop out of the table next to Ranjit’s chair, while the one named Jeannie was approaching with stethoscope, blood pressure machine, and several other instruments at the ready.
Ranjit didn’t protest. He let the doctor poke and prod and listen as much as she liked while he himself slowly and clumsily typed out pages of a nearly six-page manuscript, pausing every couple of lines to ask Gamini if he could find the address of the magazine called Nature for him. “Their offices are in England somewhere.” Or just to scowl at the keyboard until memory at last told him what the next line should be. It was a slow process, but when Gamini ventured to ask him if he wanted anything to eat, Ranjit ferociously and unarguably told him to shut up. “Just give me ten minutes,” he demanded. “Oh, maybe half an hour at the most. I can’t stop now.”
It wasn’t ten minutes, of course. Wasn’t half an hour, either. It was well over an hour before Ranjit looked up, sighed, and said to Gamini, “I need to check everything, so I’d better send a copy to your house. Tell me your e-mail address.”
And when he had typed that in, he at last pushed the icon marked send
and then sat back.
“Thanks,” he said. “I’m sorry to have been such a pest, but it was pretty important to me. Ever since I figured it out, five or six months ago, I’ve been terrified that I might forget some part of it before I could get it refereed.” He paused, suddenly licking his lips. “And one other thing. For a long time I’ve been thinking about real food. Do you have, say, any fresh fruit juice on this plane, any kind? And maybe something like a ham sandwich, or maybe a couple of scrambled eggs?”
16
HOMEGOING
Gamini refused to listen to any talk of American breakfasts but simply signaled to the flight attendants. Who produced a fine Sri Lankan meal—string hoppers of woven rice, a rich curry of meat and potatoes, and a plate of poppadoms—causing Ranjit’s eyes to bulge in wonder. “Tell me, Gamini,” he ordered, already chewing, “when did you get to be God? Isn’t this an American plane?”
Gamini, sipping a cup of tea that had come from the fields around Kandy, shook his head. “It’s a United Nations plane,” he said, “which happens to have an American crew, only it’s not on either UN or U.S. business right now. We just borrowed it to go after you.”
“And ‘we’ is—?”
Gamini shook his head again, grinning. “Can’t tell you, or anyway not right now. Pity. I knew you’d be interested, and as a matter of fact, I was considering asking you if you wanted to join us, when you went off on your little cruise.”
Ranjit didn’t put his spoon down, but he held it motionless while he gave his friend a long and not entirely friendly look. “You’re telling me that you’re such an important person that you can just borrow a plane like this to run your errands for you?”