The Last Theorem - Arthur Charles Clarke [108]
“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