Silke didn’t push his hand away. Two red spots appeared on her cheeks. “Too bad. She cares for you.”
“Listen. I’m not attracted to her. I’d rather sleep with a Gila monster. She’s fine as a friend, but stop pushing her on me.”
Silke looked behind him, her eyebrows up high, so he turned around, sloshing the whiskey in his hand. Of course Carleen had come downstairs just at that instant. Her tartan robe was hanging open and she was wearing just underpants. Elliott’s eyes went to her pale rib cage, her short, skinny legs; he couldn’t help himself.
Carleen made a strangled sound and turned and ran. Rolling her eyes at him, Silke went after her.
Elliott drank down the whiskey. He would sleep on the lumpy couch and leave early. Silke did not come back down. It seemed his chance had fled, and just for a minute there, when he said, “Immortality,” he could have said, “You,” and Silke might still be talking to him. Now Carleen was mad and complications would ensue.
All external thoughts fled, and he fell gratefully into the rutted byways of his theories, like a junkie who knows it’s dangerous but can’t fight it anymore.
Back, back to Cantor’s Continuum. The central difficulty lies at the intersection of linguistics and math analysis, he thought, the intersection of what is discrete, like the integers, and what is continuous, like infinity. What is that intersection? Where is that intersection?
The Greeks had such a horror of infinity, and it still afflicted number theory, this need to make the infinite finite, these tortured reciprocals, these sequences that lead to the infinitely small, this strange reversal of the kingly truth…
The primes are discrete, but extend into infinity like any set of integers. They are discrete in some qualitatively different way. The integers are at equal intervals from each other by definition. What if I make a number line putting the primes at equal intervals… how is that function constructed… no damn imaginary numbers, not even reciprocals to make the series converge, forget the zeta function, I’ll invent my own…
“ Wakefield!” Whiskey-tinged breath blew into his face. He lay on his back on the couch, his neck at an impossible angle. His eyes refused to open. It was very late, or maybe very early?
“You have to go home. Carleen can’t see you here in the morning.”
He reached up and around Silke’s body, finding her waist. He drew her down to him.
“ Wakefield, no…”
“It’s you. It’ll always be you.”
“Stop!”
“You, not immortality.” He squeezed tighter. He loved holding warm, soft Silke against him. He felt hot tears on his face. Hers or his? She stopped struggling and lay exhausted on top of him.
“Please,” he said. “Just this once. I need you so much.”
“Idiot,” she said. “No.”
“Then kiss me. That’s all I ask. Silke, I need you more than he does. Please.” His lips already lay against her cheek. She turned her head slightly and her mouth caught his. He drank her in.
His arms relaxed and she rolled off the couch onto the floor and went away. He turned onto his side and went back to sleep.
Sun came through the curtains. “Old man,” Raj was saying, and shaking him none too gently. “Up you go.”
“What time is it?” Elliott mumbled.
“Eight-thirty.”
“I have class at nine.”
“The bathroom is clear. The women are upstairs, but the atmosphere up there is what my mother would describe as overspiced. Get up.”
Elliott borrowed Raj’s toothbrush and splashed cold water on his face.
Raj awaited him in the kitchen. “Drink.”
Elliott drank the hot coffee, Raj watching him curiously. “What happened last night?” Raj asked.
“Nothing. I dreamed about a function to factor large numbers. Over two hundred digits.”
“Only God can do that.”
“The discrete has to be made continuous first. Cantor was close. Grothendieck…”
“Later. You have your car keys?”
Elliott felt them jingle in his pocket.
“Go.”
“I’m not sure I’ll ever come back,” Elliott said. “I think I’m going to lose all this.”
After that Elliott worked on the Riemann Hypothesis, staying at his apartment.
He ate his cold cereal and drank a lot of coffee. He sat at his kitchen table twisting bits of paper into dough.
His father called claiming to be fine, always a bad sign. He only talked of his health when he’d had an episode, but Elliott’s probing yielded no further information about any deterioration in his condition. Every few days Raj came by. Once he brought lentils and rice in a big pot. Elliott lived on that for a week, spooning portions into a bowl each night, no longer caring that it was cold food.
The problem with Raj was that he didn’t love math enough, not like Elliott did. Raj didn’t need math. As a result-his loss-he’d never be an immortal. And Silke… Elliott was glad now that she had rejected him; what a huge distraction.
With a few thousand in the bank, his tuition and rent paid, he closed the blinds. He stopped answering his phone.
Raj brought Professor Braun to see Elliott. Braun had contributed several original papers on differentiable manifolds before the age of twenty-two, and at the age of twenty-nine had been made the youngest full professor in the history of MIT. He taught the advanced number-theory course.
“You’re wet,” Elliott said as Raj and Braun took off their jackets and hung them on chairs.
“It’s pouring outside,” Braun said. “Didn’t you notice?”
“Did you come to see my work? Because it’s not ready yet. You know Erdös’s proof that a prime can always be found between an integer and its double? It has to do with the number One. The Stick. One is an integer. I don’t care if you won’t call it a prime number, you have to admit it’s an integer. Why doesn’t the proof work with One? There is no prime between One and Two. How can it be said that Erdös proved anything? Have you thought about this problem?”
“We want to take you out to lunch,” Raj said. “And Professor Braun wants you to return to class.”
“Wait, I want you to think about it. The way there’s no integer between One and Two, but where the prime should be, there is one-half. What does that remind you of, Raj?”
“Real part one-half,” Raj said softly. “Riemann’s critical line.”
“Yes! Yes! The Riemann Hypothesis! There’s a link, but I haven’t been able to prove it. Shall we talk about that?”
“We’ll talk about anything you want if you’ll come have a meal with us,” Dr. Braun said.
“Okay. I’m out of food anyway.”
They walked out into the rain, to a grubby pizza place on Mass Avenue. Drinking beer fast, eating a large salad, Elliott talked about his new suspicion that the Riemann Hypothesis was undecidable, unprovable either way. He talked about Cantor, about the discrete integers and the continuum. He explained how he had tried the algebraic approach through finite fields; how he had used Cramer’s model, treating it as a perturbation problem, trying to get a set of wobble frequencies. He talked about Sarnak and Wiles and Bump.
But mostly he talked about Cantor, the master of infinity. “I need a math that will operate with divergent series,” he explained, “comfortable with infinity. That happens if you permit division by zero.”
Raj groaned.
“Yeah, that again,” Elliott said.
They pumped him, or did they? Maybe they were just curious. He couldn’t tell. They wanted to know about his work on factoring large numbers, and he told them he had been trying a variation on the Pollard-Strassen method on his computer. He told them that Dixon ’s method using the quadratic sieve was inferior. He talked, because he hadn’t talked for so long, he had no choice. He knew he had no perspective anymore, but drunk or not, he didn’t go into detail.
He didn’t talk about his new idea, for a function that seemed to pred
ict the location of large primes. He had to work alone. The function had some flaw he couldn’t find yet, some error in the calcs. If his idea got around the campus, it might be stolen, published half-baked in some journal under someone else’s name.
He promised Dr. Braun that he would return to class the next morning. The professor gave him some class materials and said he’d pass Elliott if he turned in some work. Raj kept talking about eating better. The two of them and the other people in the restaurant seemed to be talking in some other universe, their minds occupied with all the wrong things.
“Carleen moved out of Everett Street,” Raj announced. “She claims she’s forgiven you, but she won’t hang with us anymore.”
“And Silke?”
“Did Silke have something to forgive you for?”
“Are you and Silke still taking trips?”
Raj cast a glance at Braun. “Sure,” he said. “You’re always welcome to join us again.”
“When I finish my work.”
“Do you sleep at night?” Professor Braun said, his blue eyes intent.
“I’ll sleep when I’m dead.”
“How old are you?”
“Twenty,” Elliott said. “How old are you?”
“Forty-two.”
“What have you discovered lately?”
“Don’t,” Raj said.
“You have to watch your health. I don’t like the look in your eye,” the professor went on.
“I don’t give a shit about my health.”
“Remember what happened to Cantor?”
Elliott thought of Cantor, alone in wartime at the asylum in Halle, begging his daughter to bring him home. She wouldn’t. He had always wondered how she put it: “Sorry, Father, but you’re too much trouble,” she must have said. Cantor had died swiftly after that.
“Don’t squander it,” the professor repeated, pushing the last piece of pizza away from himself. “ Wakefield, you have to pace yourself. It’s a question of balance. You should see people. Your friends are worried.”
“No, no,” Elliott said, draining his mug. “Accomplishment requires sacrifice, even pain. You have to set priorities. Right now, my work comes first. Friends, family, romps in the park, children, lovers, these things have to wait.” He stood up, carefully placing his chair under the table. “Thanks for the vitamins. I need to go.” He felt pressure to flee from their slowness and their compromises before they infected him.
“See you tomorrow, then,” said Dr. Braun.
“See you.” He gave them a friendly wave. He could afford the ten seconds it took to make them happy.
He didn’t go back to class the next day, or the next.
***
A week later, at four in the morning, Elliott finally fell into sleep, or something like sleep. He had been taking No-Doz and drinking a lot of coffee and hadn’t slept for a couple of days. He should have slept too hard for dreams to reach him.
But one did. It was early in the morning, he would remember later.
An angel appeared, her wings outspread, calling him to her and clasping him tight. He wasn’t afraid; he felt that he was already dead, and nothing worse could happen to him. In a way, he felt that this was his reward for dying. Maybe the angel was his mother.
They flew to a place at the dim limits. Here he found waves, rhythms, measures he had never known existed. All answers were here.
The angel generated a zeta landscape, infinitely dense. Even so, the mountains they flew over were porous, one or two stretching up into infinity. Between the mountains he saw deep holes, the points spiraling inward. The beauty of it made him weep.
They moved down to sea level and flew along the cross-section. The angel pointed. Then he saw it breaking through below: a huge prime, lonely and beautiful, its gray back shiny with spray. It had lost its way and was moving away from the other primes, spuming and slapping their tails in the limitless distance.
They moved into higher densities, Elliott now carrying his black notebook to write it all down. This rolling sequence, which looked like a field in Wisconsin, converged on both zero and infinity at the same time.
He looked at the angel, and the angel nodded its blazing head.
He didn’t really want to do it, but he felt he had to. He did the unthinkable. He divided by zero.
The white sky split and the rules gave way to random crime. The universe compressed into its basic reality, the four numbers surrounded by their clouds of probabilities. But Zero, staggering, was vanishing into the mist. One was a hard black branch beating the other integers. Two, the cop in blue, struggled to keep order, but it was outgunned by Three, red and bursting, rampaging all through the set.
The Three destroyed adelic space and time. Elliott, horrified, had to watch the gruesome factoring of the prime.
His angel faded away, leaving him to drift all alone in this disintegrating universe. It would all collapse into pure theory soon.
He became very frightened. Flying low across the sea, he found a hidden crater a few fathoms beneath: the square root of minus one, a geyser of fresh water rushing out of it into the saltiness.
He dove into the deep cold system.
13
TOO MANY COLLEGES, TOO MANY BROWNS. Nina recalled a math instructor at Lake Tahoe Community College she had helped with a contract problem once, Mick McGregor. Mick had his math doctorate, but had lost his first job at UC Berkeley for reasons he had never told her. Luckily for her today, he had landed right around the corner.
High noon on Thursday at the LTCC campus, and the place wasn’t exactly hopping although school was in session. Nina parked among shady pines and walked to the Administration Building. An art exhibit was going on inside, student sculptures propped amid the seats in the high-ceilinged reception area, paintings on the wall, but few students circulated. The building was brand-new and even the carpeting still looked fresh and welcoming.
At the registrar’s window she was directed to McGregor’s office, but she found him outside the building talking to a student.
“Uh-oh,” he said when he recognized her. He might as well have said, Here comes trouble. She often called forth that reaction, a hazard of her trade.
“Everything’s fine,” she assured him. “I’ve come to consult you about something I’m working on.” The student wandered away and they entered McGregor’s office, which was as neat as a marine’s cot. Nina looked around approvingly at the orderly books and papers and photos of McGregor with his family and students.
“Long time,” McGregor said. “I still wake up at night remembering how worried I was in those days that I’d lose the case. No offense, but you’re always in the dreams.”
“For Pete’s sake, Mick. We won the case. Can’t you rehabilitate me? I hate to think I’m part of somebody’s nightmares.” Nina was only half-joking.
“Let’s start over,” Mick said. “Let me think of you as a very pretty lady who dropped in on me unannounced for a chat about nothing much.” He was still young, with reddish hair, wearing jeans and a purple shirt with a white undershirt showing at the neck, his hands freckled, his manner ironic.
“I’d like to ask you about some math students, at least I think they may be math students, who I need to locate. They may be in Boston.”
“I was raised in Lawrence, Massachusetts. But you knew that.”
“I did know that.”
“What do you want to know?”
“I want to find them. I know three things: One or more has or had a professor named Brown; they flew from here to Boston; and one of them bought two books at Sierra Books here at Tahoe while they were visiting.”
“The plane was flying at the rate of four hundred miles an hour, and the student read one-fourth of the first book on the flight. Elementary. Want me to write out the proof for you?”
Nina handed him the names of the books. “I want to know what kind of math this is, and what level.”
McGregor read the names. “Ooh. Somebody’s into the Riemann Hy
pothesis. And Cantor’s Continuum Hypothesis. I’ve seen these books at Sierra Books gathering dust. It’s a wonder they carried them.”
“What can you tell me about these-these hypotheses?”
“Analytic number theory. A fancy word for arithmetic, but do not be misled. This is graduate-level stuff, very sexy math, very deep. Hardly anybody is working on it. The universities want combinatorics people and physicists and topologists these days. Students who read books on these topics on vacation are going to be obsessed with the hypotheses. For them it’s the closest thing to fun.”
“It sure doesn’t sound like fun to me. So would you think this student was working on a graduate degree?”
“Could be an advanced undergraduate. These books concern the most famous problems in math. Some people think figuring out whether the Riemann Hypothesis is true or not is the biggest mystery in the universe, bar nothing. It has to do with the prime numbers. You know? Prime numbers?”
“I remember that they’re the numbers that you can’t divide anything into,” Nina said. “Am I saying it right?”
“Sure. They’re the basic building blocks all the other numbers consist of. But they have a devilish aspect. They appear randomly on the positive number line. There’s no satisfactory algorithm that identifies them in sequence, and with large numbers it’s almost impossible to find the factors and determine if the numbers are prime or not.”
“So what?” Nina said.
“So what, you ask. Well, if we can’t find a formula to predict such a basic and crucial number sequence, we look like clowns, and the whole orderly system of mathematics we’ve built up over twenty-five hundred years looks like a pile of shit,” McGregor said. “It’s the black hole at the center of this area of human knowledge. We don’t even know what the fuck prime numbers are. Maybe they’re aliens from outer space sent to drive us crazy.”
“Oh. Aliens. Sure.”
“Riemann gave us a big clue about the behavior of the primes a hundred fifty years ago, but nobody has managed to take full advantage of it. Until his hypothesis is proved, we’re all a bunch of buffoons. Same with Cantor’s work on infinity. Until we figure out what to do with series that diverge, we may as well admit our whole mathematics system is a joke.”
Case of Lies Page 13