The Return of the Black Widowers (19 page)

"Actually," said Halsted, "it's very simple. Back in 1742, I think, a Russian mathematician, Christian Goldbach, stated that he believed every even number greater than 2 could be written as the sum of two primes, where a prime is any number that can't be divided evenly by any other number but itself and 1. For instance, 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 10 = 3 + 7; 12 = 5 + 7; and so on, as far as you want to go."

Gonzalo said, "So what's the big deal?"

"Goldbach wasn't able to prove it. And in the two hundred and something years since his time, neither has anyone else. The greatest mathematicians haven't been able to show that it's true."

Gonzalo said, "So?"

Halsted said patiently, "Every even number that has ever been checked always works out to be the sum of two primes. They've gone awfully high and mathematicians are convinced the conjecture is true—but no one can
prove
it."

Gonzalo said, "If they can't find any exceptions, doesn't
that
prove it?"

"No, because there are always numbers higher than the highest we've checked, and besides we don't know all the prime numbers and can't, and the higher we go, then the harder it is to tell whether a particular number is prime or not. What is needed is a
general
proof that tells us we don't have to look for exceptions because there just aren't any. It bothers mathematicians that a problem can be stated so simply and seems to work out, too, and yet that it can't be proved."

Trumbull had been nodding his head. "All right, Roger, all right. We get it. But tell me, does it
matter!
Does it really matter to anyone who isn't a mathematician whether Goldbach s conjecture is true or not; whether there are any exceptions or not?"

"No," said Halsted. "Not to anyone who isn't a mathematician; but to anyone who is and who manages either to prove or disprove Goldbach s conjecture, there is an immediate and permanent niche in the mathematical hall of fame."

Trumbull shrugged. "There you are. What Pochik's really doing is of great importance. I'm not sure whether it's for the Department of Defense, the Department of Energy, NASA, or what, but it's vital. What
he's
interested in, however, is Goldbach's conjecture, and for that he's been using a computer."

"To try higher numbers?" asked Gonzalo.

Halsted said promptly, "No, that would do no good. These days, though, you can use computers on some pretty recalcitrant
problems. It doesn't yield an elegant solution, but it is a solution. If you can reduce a problem to a finite number of possible situations—say, a million—you can program a computer to try every one of them. If every one of them checks out as it's supposed to, then you have your proof. They recently solved the four-color mapping problem that way; a problem as well known and as recalcitrant as Goldbach's conjecture."

"Good," said Trumbull, "then that's what Pochik's been doing. Apparently, he had worked out the solution to a particular lemma. Now what's a lemma?"

Halsted said, "It's a partway solution. If you're climbing a mountain peak and you set up stations at various levels, the lemmas are analogous to those stations and the solution to the mountain peak."

"If he solves the lemma, will he solve the conjecture?"

"Not necessarily,” said Halsted, "any more than you'll climb the mountain if you reach a particular station on the slopes. But if you
don't
solve the lemma, you're not likely to solve the problem, at least not from that direction."

"All right, then," said Trumbull, sitting back. "Well, Sandino came up with the lemma first and sent it in for publication."

Drake was bent over the table, listening closely. He said, "Tough luck for Pochik."

Trumbull said, "Except that Pochik says it wasn't luck. He claims Sandino doesn't have the brains for it and couldn't have taken the steps he did independently; that it is asking too much of coincidence."

Drake said, “That’s a serious charge. Has Pochik got any evidence?"

"No, of course not. The only way that Sandino could have stolen it from Pochik would have been to tap the computer for Pochik's data and Pochik himself says Sandino couldn't have done that."

"Why not?" said Avalon.

"Because," said Trumbull, "Pochik used a code word. The code word has to be used to alert the computer to a particular person's
questioning. Without that code word, everything that went in
with
the code is safely locked away."

Avalon said, "It could be that Sandino learned the code word."

"Pochik says that is impossible," said Trumbull. "He was afraid of theft, particularly with respect to Sandino, and he never wrote down the code word, never used it except when he was alone in the room. What's more, he used one that was fourteen letters long, he says. Millions of trillions of possibilities, he says. No one could have guessed it, he says."

Rubin said, "What does Sandino say?"

"He says he worked it out himself. He rejects the claim of theft as the ravings of a madman. Frankly, one could argue that he's right."

Drake said, "Well, let's consider. Sandino is a good mathematician and he's innocent till proven guilty. Pochik has nothing to support his claim and Pochik actually denies that Sandino could possibly have gotten the code word, which is the only way the theft could possibly have taken place. I think Pochik has to be wrong and Sandino right."

Trumbull said, "I
said
one could argue that Sandino s right, but the point is that Pochik won't work. He's sulking in his room and reading poetry and he says he will never work again. He says Sandino has robbed him of his immortality and life means nothing to him without it."

Gonzalo said, "If you need this guy so badly can you talk Sandino into letting him have his lemma?"

"Sandino won't make the sacrifice and we can't make him unless we have reason to think that fraud was involved. If we get any evidence to that effect we can lean on him hard enough to squash him flat.—But now listen, I think it's possible Sandino
did
steal it."

Avalon said, "How?"

"By getting the code word. If I knew what the code word was, I'm sure I could figure out a logical way in which Sandino could
have found it out or guessed it. Pochik, however simply won't let me have the code word. He shrieked at me when I asked. I explained why, but he said it was impossible. He said Sandino did it some other way—but there is no other way."

Avalon said, "Pochik wants an interpretation but he won't tell you the dream, and you have to figure out the dream first and then get the interpretation."

"Exactly! Like the Chaldean wise men."

"What are you going to do?"

"I'm going to try to do what Sandino must have done. I'm going to try to figure out what the fourteen-letter code word was and present it to Pochik. If I'm right, then it will be clear that what I could do, Sandino could do, and that the lemma was very likely stolen."

There was a silence around the table and then Gonzalo said, "Do you think you can do it, Tom?"

"I don't think so. That's why I've brought the problem here. I want us all to try. I told Pochik I would call him before 10:30 P.M. tonight"—Trumbull looked at his watch—"with the code word just to show him it
could
be broken. I presume he's waiting at the phone."

Avalon said, "And if we don't get it?"

"Then we have no reasonable way of supposing the lemma was stolen and no really ethical way of trying to force it away from Sandino. But at least we'll be no worse off."

Avalon said, "Then you go first. You've clearly been thinking about it longer than we have, and it's your line of work."

Trumbull cleared his throat. "All right. My reasoning is that if Pochik doesn't write the thing down, then he's got to remember it. There are some people with trick memories and such a talent is fairly common among mathematicians. However, even great mathematicians don't always have the ability to remember long strings of disjointed symbols and, upon questioning of his coworkers, it would seem quite certain that Pochik's memory is
an ordinary one. He can't rely on being able to remember the code unless it's easy to remember.

"That would limit it to some common phrase or some regular progression that you couldn't possibly forget. Suppose it were ALBERT EINSTEIN, for instance. That's fourteen letters and there would be no fear of forgetting it. Or SIR ISAAC NEWTON, or ABCDEFGHIJKLMN, or, for that matter, NMLKJIHGFEDCBA. If Pochik tried something like this, it could be that Sandino tried various obvious combinations and one of them worked."

Drake said, "If that's true, then we haven't a prayer of solving the problem. Sandino might have tried any number of different possibilities over a period of months. One of them finally worked. If he got it by hit-and-miss over a long time, we have no chance in getting the right one in an hour and a half, without even trying any of them on the computer."

"There's that, of course," said Trumbull, "and it may well be that Sandino had been working on the problem for months. Sandino pulled the waiter routine on Pochik last June, and Pochik, out of his mind, screamed at him that he would show him when his proof was ready. Sandino may have put this together with Pochik s frequent use of the computer and gotten to work. He may have had months, at that."

"Did Pochik say something on that occasion that gave the code word away?" asked Avalon.

"Pochik swears all he said was 'I'll show you when the proof is ready,' but who knows? Would Pochik remember his own exact words when he was beside himself?"

Halsted said, "I'm surprised that Pochik didn't try to beat up this Sandino."

Trumbull said, "You wouldn't be surprised if you knew them. Sandino is built like a football player and Pochik weighs 110 pounds with his clothes on."

Gonzalo said, suddenly, "What's this guy's first name?"
Trumbull said, "Vladimir."

Gonzalo paused a while, with all eyes upon him, and then he said, "I knew it. VLADIMIR POCHIK has fourteen letters. He used his own name."

Rubin said, "Ridiculous. It would be the first combination anyone would try."

"Sure, the purloined letter bit. It would be so obvious that no one would think to use it. Ask him."

Trumbull shook his head. "No, I can't believe he'd use that."

Rubin said, thoughtfully, "Did you say he was sitting in his room reading poetry?"

"Yes."

"Is that a passion of his? Poetry? I thought you said that outside mathematics he was not particularly educated."

Trumbull said, sarcastically, "You don't have to be a Ph.D. to read poetry."

Avalon said, mournfully, "You would have to be an idiot to read modern poetry."

"That's a point," said Rubin. "Does Pochik read contemporary poetry?"

Trumbull said, "It never occurred to me to ask. When I visited him, he was reading from a book of Wordsworth's poetry, but that's all I can say."

"That's enough," said Rubin. "If he likes Wordsworth then he doesn't like contemporary poetry. No one can read that fuddy-duddy for fun and like the stuff they turn out these days."

"So? What difference does it make?" asked Trumbull.

"The older poetry with its rhyme and rhythm is easy to remember and it could make for code words. The code word could be a fourteen-letter passage from one of Wordsworth's poems, possibly a common one: LONELY AS A CLOUD has fourteen letters. Or any fourteen-letter combinations from such lines as 'The child is father of the man' or 'trailing clouds of glory' or 'Milton! thou shouldst be living at this hour.'—Or maybe from some other poet of the type."

Avalon said, "Even if we restrict ourselves to passages from the classic and romantic poets, that's a huge field to guess from."

Drake said, "I repeat. It's an impossible task. We don't have the time to try them all. And we can’t tell one from another without trying."

Halsted said, "It's even more impossible than you think, Jim. I don't think the code word was in English words."

Trumbull said, frowning, "You mean he used his native language?"

"No, I mean he used a random collection of letters. You say that Pochik said the code word was unbreakable because there were millions of trillions of possibilities in a fourteen-letter combination. Well, suppose that the first letter could be any of the twenty-six, and the second letter could be any of the twenty-six, and the third letter, and so on. In that case the total number of combinations would be 26 x 26 x 26, and so on. You would have to get the product of fourteen 26's multiplied together and the result would be"—he took out his pocket calculator and manipulated it for a while—"about 64 million trillion different possibilities.