“Sounds complicated,” his mother commented. “Also, that part about selecting them at random by computer? That was suggested thirty years or so ago by a science-fiction writer.”
Joris nodded. “They have all the best ideas, don’t they? Anyway, the system can’t work until they get their communications back—at least another month or two, I think. Maybe by then we’ll understand it.”
After dinner the proud parents had to show Joris how well their infant could swim, and Mevrouw insisted that Joris go to bed when Tashy did. Since the last time he’d been in a bed, he had flown halfway around the world, and it was time he got some rest!
So there wasn’t any chance to ask for Joris’s help then, either. When both Natasha and his wife were sound asleep, Ranjit fretfully flipped on the news, sitting in their dressing room, volume too low to disturb the sleepers. The Security Council had issued a whole new bundle of stern warnings to countries that were engaging in, or seemed to be on the brink of, one of those brushfire wars; Silent Thunder was not mentioned but, Ranjit had no doubt, was present in the calculations of all the belligerents. It was possible, Ranjit told himself, that he had made a mistake in turning Gamini’s offer down. Pax per Fidem had every appearance of being where the action was, while Colombo did not.
Irritated, he turned off the news. He thought he might as well get some sleep and perhaps get a word with Joris first thing in the morning, before Joris was off again on his way to the terminal’s construction site.
But there was a faint sound of music coming from somewhere.
Ranjit pulled on a robe and investigated. There, on the balcony overlooking the gardens, Joris sat, sipping a tall drink and gazing at the moon while a radio softly played. When he saw Ranjit peering at him, he gave him a faintly embarrassed grin. “You caught me. I was just thinking where I’d like to land up there, oh, maybe five or six years from now, when the Skyhook’s operational and I can get there. Mare Tranquillitatis, or Crisium, or maybe something on the far side, just to show off. Sit down, Ranjit. Would you care for a nightcap?”
Ranjit certainly would, and Joris had the fixings all ready for them. As he accepted the glass, Ranjit nodded toward the moon, nearly full, bright enough, almost, to read by. “Do you really think you’ll be able to do that?” he asked.
“I don’t think it; I guarantee it,” Vorhulst promised. “Maybe it’ll take a little longer for your average man in the street to buy a ticket. Not me. I’m an executive in the program, and rank has its privileges.” He took note of a faintly quizzical expression on Ranjit’s face. “What is it? You never expected me to take advantage of a position to get something I wanted? Well, for most things I wouldn’t. But space travel is special. If the only way to get to the moon would be by robbing banks to finance the trip, I’d rob banks.”
Ranjit shook his head. “I wish I liked my job as much as you like yours,” he said, feeling a tiny stab of what he could recognize only as jealousy.
Dr. Vorhulst gave his former student a considering look. “Have a refill,” he offered. And then, while he was mixing one, he said, “And while we’re here, how would you like to tell me how you and the university are getting along?”
Ranjit would have, of course, liked nothing better. It didn’t take long for him to unload his problems onto his former teacher, and not as long as that for Joris Vorhulst to get the picture. “So,” he said thoughtfully, again replenishing their glasses, “let’s get back to basics. You don’t have any trouble filling a class, do you?”
Ranjit shook his head. “For the first seminar, they had a waiting list thirty or forty people long that couldn’t get in.”
“So then, why do people sign up for a class with you? It isn’t because you’re a great teacher—even if you were, they wouldn’t have had any chance to find that out. It isn’t because abstruse mathematics has suddenly got popular. No, Ranjit, the thing that pulls them in is you yourself, and how you plugged away at that problem for all those years. Why don’t you teach them to do as you did?”
“Tried it,” Ranjit said glumly. “They said they’d heard me lecture on that already. They wanted something new.”
“All right,” Joris said, “then why don’t you show how someone else solved a problem like that, step by step….”
Ranjit looked at him with dawning hope. “Huh,” he said. “Yes, maybe. I know a lot about the way Sophie Germain tried to do Fermat herself—didn’t succeed, of course, except partially.”
“Fine,” Joris said with satisfaction, but Ranjit was still thinking.
“Or, wait a minute,” he was saying, suddenly excited, “do you know what I could do? I could take one of the grand old problems that nobody has solved—say, Euler’s reworking of the Goldbach conjecture; you can explain that in words of one syllable that anybody can understand, though nobody’s ever been able to produce a proof. What Goldbach proposed—”
Joris’s hand was raised. “Please don’t explain this Goldbach conjecture to me. But, yes, that sounds good. You could do it as a sort of class project. Everybody working on it together, the students and you as well. Who knows? Maybe you could even solve the thing!”
That produced an actual laugh from Ranjit. “That would be the day! But it doesn’t matter; the students would at least get a feeling of what it takes to solve a big problem, and that ought to hold their interest.” He nodded to himself, pleased. “I’ll try it! But it’s getting late and you have to get up in the morning, so thanks, but let’s call it a night.”
“We’d better do that before my mother catches me still up,” Vorhulst agreed. “But there’s something else I wanted to talk to you about, Ranjit.”
Ranjit, on the point of getting up to leave, paused with his hands on the arms of his chair, ready to lift. “Oh?”
“I’ve been thinking about that committee you were invited to go to work for at old Peace Through Transparency. It occurs to me that maybe we need something like it for the ladder. Famous people keeping an eye on what we’re doing and now and then telling the world about it. Famous people like you, Ranjit. Do you think you might consider—?”
Ranjit didn’t let him finish. “Whatever you’re asking,” he said, “the answer is yes. After all, you’ve just saved my life!”
And “yes” it was…and years later Ranjit considered with wonder how that simple single word had changed his life.
Some light-years away the lives of the 140,000 One Point Fives in the Earth-depopulation fleet were also on the verge of a major change.
By the calculations of their Machine-Stored navigators the flotilla was within thirteen Earth years of their assault on the doomed human race. That was a meaningful point for the One Point Fives. It meant it was time for an important action to be taken.
So all over the fleet, in every corner of every ship, specially trained technical crews were checking every instrument or machine that was working, and turning most of them off. Basic drive, off; that meant the fleet was now simply drifting toward Earth—though already at such a great velocity that, under Einstein’s laws, further acceleration was very difficult and very nearly pointless. Airborne waste filters: off. So beginning at once the exhalations of the One Point Fives themselves would begin to contaminate the air they breathed. Power pack chargers, off. Search beams, off. The instruments that monitored the running of all the machinery that couldn’t be turned off even briefly—off.
Suddenly the armada of the One Point Fives was no longer a hard-driving fleet of warcraft aiming at a point of conflict; it had suddenly become a collection of derelicts, almost powerless and approaching the point where one ship might drift into another. The fleet could not maintain that condition for long.
The One Point Fives, however, didn’t need it to be for long. As soon as the last crew reported that everything that could be shut down was shut down, the One Point Fives began slipping out of the last vestiges of their protective armor and life supports. Then began the wildest orgy of sexual activity any One Point Five could imagine.
For about an hour.
Then the pallid creatures that were the organic One Point Fives hurriedly clambered back into their armor. In each ship the technical crews hastily retraced their steps, turning back on everything they had turned off, and the orgy was over.
Why did the One Point Fives do that?
For a reason that most human beings would have understood very easily. One Point Fives, whether armored or stripped to their wasted little organic bodies, did not look in the least like any humans, but they had some traits in common. No One Point Five wanted to die without leaving a live descendant to take his place. In the struggle that lay ahead there was a definite and nonzero chance that some or all of them would die. So in that collective mating, many—with luck, maybe most—of the females would become pregnant. The fifteen Earth years before that final conflict was the minimum time it would take for them to deliver their wretched little newborns to the nursery machines, and then for the infants to grow and mature to the stage of puberty.
With that knowledge, their parents could afford to launch the attack.
Of course, no human being knew this, so all nine billion of them went right on with their usual daily tasks, none of them knowing that, from that day forward, their own newborns could only expect to experience the first inklings of sexual maturity before being wiped from the face of the earth.
29
BURGEONING HOPES
In the event, Ranjit did not begin his next seminar with Goldbach’s conjecture. Myra had a different suggestion, and he had learned to listen to Myra.
The first day he faced the class, he spent most of the opening hour on housekeeping matters—answering questions on his testing and grading policies, announcing what days of class would be skipped for higher-echelon reasons, getting to begin to know some of the students. Then he asked, “How many of you know what a prime number is?”
Nearly every hand in the room went up. Half a dozen of the students didn’t wait to be recognized but called out one version or another of the definition: a number that can be divided, without a remainder, only by one and itself.
It was a promising beginning. “Very good,” Ranjit told them. “So two is a prime number and three is a prime number, but four can be divided not only by itself and by one but also by two. It is not, therefore, a prime number. Next question: How do you generate prime numbers?”
There was stirring in the classroom, but no hands immediately arose. Ranjit grinned at his students. “That’s a hard question, isn’t it? There are a bunch of shortcuts that people have suggested, many of them requiring large computers. But the one way that requires nothing but a brain, hand, and something to write with—but is guaranteed to generate every prime number there is up to any limit you care to set—is something called the sieve of Eratosthenes. Anybody can use the sieve. Anybody with a lot of time on his hands, that is.”
He turned and began writing a line of numbers on the whiteboard, everything from one to twenty. As he was writing, he said, “There’s a little mnemonic poem to help you remember it:
Strike the twos and strike the threes,
The sieve of Eratosthenes.
When the multiples sublime,
The numbers that are left are prime.
“That’s the way it works,” he went on. “Look at the list of numbers. Ignore the one; there’s a sort of gentlemen’s agreement among number theorists to pretend that the one doesn’t belong there and shouldn’t be called a prime, because just about every theorem about number theory goes all wonky if it includes the one. So the first number on the list is two. Now you go along the list and strike out every even number. That is, every number divisible by two, after the two itself—the four, six, eight, and so on.” He did that. “So now the smallest number left, after the original two and the one that we’re pretending never existed, is the three, so we strike out the nine and every later number left on the board that is divisible by three. So that leaves us with the two, the three, the five, the seven, and the eleven, and so on. And now you’ve generated a list of the first prime numbers.
“Now, we’ve only gone up to twenty because my hand gets tired when I write long lists, but the sieve works for any number of digits. If you were to write down the first ninety thousand numbers or so—I mean everything from one to around ninety thousand—your last surviving number would be the one-thousandth prime, and you would have written every prime before that as well.
“Now”—Ranjit glanced at the wall clock, as he had seen so many of his own teachers do—“because these are three-hour sessions, I’m declaring a ten-minute intermission now. Stretch your legs, use the facilities, chat with your neighbors—whatever you like, but please be back in your seats at half-past the hour, when we’ll begin to take up the real business of the seminar.”
He didn’t wait to see them disperse but ducked quickly into the private door that led to faculty offices down the hall. He used his own facilities—pee whenever you get the chance, as, according to an urban legend, a queen of England had once counseled her subjects—and quickly called home. “How is it going?” Myra demanded.
“I don’t know,” he said honestly. “They’ve been quiet so far, but a fair number of them have put up their hands when I’ve asked questions.” He considered for a moment. “I think you could say that I’m cautiously optimistic.”
“Well,” his wife said, “I’m not. Not cautious about it, I mean. I think you’re going to knock them dead, and when you come home, we’ll celebrate.”
They were all in their seats when he returned to the podium, a minute before the big hand hit the six. A good sign, Ranjit thought hopefully, and plunged right in.
“How many prime numbers are there?” he asked, without preface.
This time the hands were slow in going up, but nearly all managed it. Ranjit pointed to a young girl in the first row. She stood and said, “I think there are an infinite number of prime numbers, sir.” But when Ranjit asked why she thought that, she hung her head and sat down again without answering.
One of the other students, male and older than the rest, called out, “It’s been proved!”
“Indeed it has,” Ranjit agreed. “If you make a list of prime numbers, no matter how many are on the list or how big the biggest of them is, there will always be other primes that aren’t on the list.
“Specifically, let’s make believe that we’re all pretty dumb about numbers and so we think that maybe the last term in that list, nineteen, is the biggest prime number that ever could be. So we make a list of all the primes smaller than nineteen—that is, two through seventeen above, and we multiply them all together. Two times three times five, et cetera. We can do this because, although we’re pretty dumb, we have a really good calculator.”
Ranjit allowed time for a few giggles to survive, then went on. “So we’ve done the multiplication and obtained a product. We then add one to it, leaving us with a number we will call N. Now, what do we know about N? We know that it might turn out to be a prime itself, because, by definition, if you divide by any of those numbers, you have one left over as a remainder. And if it happens to be a composite number, it can’t have any factor that is on that list, for the same reason.
“So we’ve proved that no matter how many primes you put in a list, there are always primes larger still that aren’t on the list, and thus the number of primes is infinite.” He paused, looking the students over. “Any of you happen to know who gave us that proof?”
No hands were raised, but around the classroom names were called out: “Gauss?” “Euler?” “Lobachevsky?” And, from the back row, “Your old pal Fermat?”
Ranjit gave them a grin. “No, not Fermat, and not any of the others you mentioned. That proof goes way back. Almost as far as Eratosthenes, but not quite. The man’s name was Euclid, and he did it somewhere around 300 B.C.”
He held up an amiably cautionary hand. “Now let me show you something else. Look at the list of prime numbers. Notice how often there are tw
o prime numbers that are consecutive odd ones. These are called prime pairs. Anyone care to guess how many prime pairs there are?”
There was a rustle of motion, but otherwise silence until some brave student called out, “An infinity?”
“Exactly,” Ranjit said. “There is an infinite number of prime pairs…and for your homework assignment you can find a proof of that.”
And so at dinner that night Ranjit was more spontaneously cheerful than Myra had seen him in some time. He informed the family, “They made jokes with me. It’s going to work!”
“Of course it is,” his wife said. “I had no doubt. Neither did Tashy.”
And indeed little Natasha, now allowed to join the grown-ups at dinner, seemed to be listening attentively from her high chair when the butler came in. “Yes, Vijay?” Mevrouw said, looking up. “You look worried. Is there a difficulty below-stairs?”
He shook his head. “Not below-stairs, madam. There was something on the news that I thought you might want to know about, though. There’s been another of those Silent Thunder attacks, in South America.”
This time it wasn’t a single nation that had been driven back to the pre-electronic age. This time there were two of them. Nowhere throughout the countries of Venezuela and Colombia did a telephone now ring, or a light go on when a switch was pressed, or a television display its picture.
So the rest of that meal was completed with little additional conversation about Ranjit’s seminar, or even about the skillful way Natasha was manipulating her spoon. The room’s own screens, never used during meals because Mevrouw thought that was barbarous, were full on now.
As with Korea, there were few scenes from inside either of the freshly subjugated countries, because the local facilities were all now blacked out. What was on the screens was a few sketchy displays of Pax per Fidem cargo planes—the kind with short takeoff and landing capabilities so they could dodge around the frozen aircraft on the runways—bringing in the same sort of troops and equipment that had poured over the border into North Korea. Mostly what was on the screens was talking heads—saying much the same things they had said about Korea—and a lot of stock footage to display the events that had brought on the current disaster.