Arthur Clarke - The Last Theorem

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:

“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 two 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.