The Last Theorem - Arthur Charles Clarke [107]
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