“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.
The twenty-first century had not been good to either of the two countries. In Venezuela it was politics, in Colombia drugs; in both countries there had been violence and frequent governmental crises, capped by the decision of the former narcotics lords to take over some of their neighbor’s now far more profitable oil business.
“Pax per Fidem took on North Korea first because it didn’t have a real friend in the world,” Ranjit told his wife. “This time they took on two countries at once because they had different friends—the U.S. has been propping Colombia up since the nineties, and Venezuela was close to both Russia and China.”
“But there’s a lot less killing going on now,” Mevrouw said thoughtfully. “I can’t feel unhappy about that.”
Myra sighed. “But do you think we’ll be better off when the whole world is run by Oceania, Eurasia, and Eastasia?” she asked.
30
BIG NEWS
When the seminar was over, no student had managed to produce a rigorous proof of the infinitude of twin primes, but then Ranjit hadn’t expected one would. Neither had Dr. Davoodbhoy. At their postseminar conference, though, he was visibly happier than before. He flourished the student comment slips at Ranjit with a grin. “Listen to these. ‘I had the feeling I wasn’t just learning how to do mathematics; I was learning what doing mathematics was all about.’ ‘Good stuff. Dr. Subramanian doesn’t treat us like children, more like we were new members of his research team.’ ‘Can I take his next seminar, too?’ And what would you say to”—he glanced again at the slip—“this young lady, Ramya Salgado?”
Ranjit looked uncomfortable. “I know who she is; she was very active in the seminar. Maybe if we needed another warm body to fill the class out.”
“Oh,” said Dr. Davoodbhoy, “I don’t think you need to worry about that. You do want to do another, don’t you? Have you thought of a subject? Maybe something like the Riemann conjecture?”
“There are proofs of that,” Ranjit reminded him.
“Some people don’t think they’re satisfactory. Anyway, there was a proof of Fermat, too—Wiles’s—and that didn’t keep you from finding a better one.”
Ranjit considered, then shook his head. “I’m afraid Riemann is too complicated for anybody but a professional mathematician to care about. How are you going to get the average college student to care about what way the zeros in the Riemann zeta function are distributed? There are better ones around. Euler’s reworking of the Goldbach conjecture, for instance. That’s pure gold. ‘All positive even integers greater than four can be expressed as the sum of two primes.’ Six is three plus three, eight is five plus three, ten is five plus five—or seven plus three, if you like that better. Anybody can understand that! Only nobody has ever proved it—yet.”
Davoodbhoy considered for one of the smaller fractions of a second, then nodded. “Go for it, Ranjit. I might even like to audit one of those sessions myself.”
As the years flowed from that point in time onward, Ranjit began to realize that he truly loved teaching. Each semester brought a new flock of eager students, and of course he had his monthly reviews of the ladder to tend to, and Natasha was growing from a young, promising girl to a slightly older girl of significant promise. If anyone in the world shared Myra’s concerns about the three Pax per Fidem sponsors’ dividing the world among them, there was little sign of it. Silent Thunder was as gentle a conquistador in South America as it had been on the Korean peninsula. The casualty list was not much longer. The problems of feeding and caring for the suddenly technology-less populations were as quickly met. The outside world observed, and discussed, and seemed to think that Pax per Fidem had done a reasonably good thing.