Fermat's Lost Theorem

Illustration by Janet Aulisio Dannheiser

When Rick Hopkins met the new math professor, he felt the sudden conviction that he’d slept with her. Trouble was, he couldn’t remember when.

The faculty lounge was abuzz with people milling about, cocktail glasses and colored napkins in hand. Marsha Selwood stood amid a small knot of professors and assistant profs near the punch bowl while Vince Gardoni, the department head, introduced her around. Tall, blonde, with high cheekbones and green eyes, she would have been the center of attention even if she hadn’t been the guest of honor. The only people in the room who weren’t fawning over her were the other women, and the dozen or so grad students who hovered over the hors d’œuvres, nervously packing away rumaki and canapes as if they were french fries. Rick didn’t recognize half of them, but he supposed their formal clothing accounted for that. Put them in baggy jeans and flannel shirts and they would suddenly become familiar again.

Maybe that was the problem with Marsha. Rick must have met her under different circumstances, and he just wasn’t making the connection.

Had he really slept with her? A moment ago he’d have sworn he had, but now he wasn’t sure. There’d been that flash of recognition, a momentary glimpse of the two of them entwined in an unfamiliar bed, but nothing after that, and now even the flash was gone. He couldn’t remember anything about the actual act, where it had been, or when. It couldn’t have been recently; he’d been away all through the holidays, and she’d been teaching halfway across the country during the fall semester. Maybe it had just been wishful thinking.

Then again, maybe not. That look on her face was unmistakable. She was waiting for him to say something, to acknowledge their special secret, and Rick had about two seconds to come up with the right response or he was dead meat.

When in doubt, go for the truth. Rick smiled at her and said, “I just had the strongest sensation of déjà vu,” putting as much spin into the words as he could without alerting Vince to the subtext.

Marsha blinked her green eyes at him and said, “Now that’s odd, so did I.”

Not to be outdone, Vince said, “You’re kidding. I did too.”

Rick and Marsha exchanged a knowing glance, then Marsha turned to Vince. “Oh really? What about?”

Three more grad students entered the lounge, two guys flanking a woman, all three of them wearing the same style suit. Gray, wide lapels, red ties. They stomped the snow from their shoes—black running shoes, it looked like—and made straight for the punch bowl. When they drew close, Rick could see that the woman’s oversized glasses had tiny winking lights around the rims, but facing inward, where she could see them better than anyone else could. Rick smiled at the odd trio and sipped his wine while Vince said, “It was really strange, but for just a second there I was sure we were holding an awards ceremony for Rick.”

Rick tried not to snort Chablis through his nose. The idea of Vince awarding him anything was laughable. Vince had barely spoken to him since their run-in with the NSA over Rick’s code-breaking algorithm. Marsha didn’t seem to realize that, though, because she asked seriously, “Oh really, what for?”

Vince laughed. “Oh, nothing much. Just solving Fermat’s last theorem.”

“That’s been done,” Rick said. “Andrew Wiles, at Princeton.”

Waving a green napkin in dismissal, Vince said, “No no, Wiles’s proof was a big, ugly, two-hundred page thing. I mean Fermat’s proof, the elegant one he didn’t bother to write down.”

Marsha’s eyes glittered with mischief. “Well, Rick,” she said, “Looks like we know what you’ll be doing for the next couple of years, eh?”

“What?” Rick frowned. Both Marsha and Vince were watching him like vultures. Like mathematicians who’d just laid down a challenge. “Oh no,” he said. “If you think I’m going to waste my time on that, you’re nuttier than Fermat.”

But as he said that, he felt a flash of insight, a tantalizing glimpse of the proof, as if he’d already solved it. The moment he focused his attention on it, though, it was gone, just like his night in the sack with Marsha.

On the way home, while he was giving a ride to Nigel, one of the other single men in the department, Nigel said, “I didn’t realize you were so popular. I had three different people ask me about you tonight.”

“Oh really?” Rick concentrated on keeping the Volkswagen from sliding on the icy streets. It had snowed on New Year’s Eve, and traffic had packed it all down before the plows could clear it away. Now they would be stuck with it for months.

“Yeah, they kept asking about your work, and what you were planning next.”

Rick groaned. “It wasn’t those bastards from the NSA again, was it? I gave up cryptography over a year ago.”

“No, I don’t think so,” Nigel said. “These guys weren’t near as slick.”

The National Security Agency people had definitely been slick. When Rick had inadvertently stepped on federal toes with his code-breaking project, they had materialized like dark wraiths. A few soft words to the dean of the college, a midnight visit to his office to clean off his hard drive—they’d even found his backup disks in the closet at home, though Rick had found no evidence of forced entry to the house. They had swept through his life in a single night, and Rick had suddenly found himself without a project. A whole year down the tubes, and strict orders from Vince to lay off cryptography.

Now he realized what tonight’s little exchange had been all about. Rick hadn’t picked a new project yet. He wasn’t doing anything exciting, and he hadn’t written any papers for over a year. Vince’s little bit about Fermat’s last theorem hadn’t been one-upmanship in the presence of a beautiful new colleague; it had been a hint. Do something or wind up teaching remedial algebra for the rest of your life.

An oncoming car seemed to materialize out of nowhere, sliding into their lane. Rick dodged to the right, narrowly missing a parked delivery van. “Jeez, where did that guy come from?” he said as the headlights swept past, but as he spoke, his mind was already leaping off on a tangent.

“Graphically,” he whispered. “I can solve it graphically.”

“What?” Nigel asked.

“Nothing,” Rick said.

He lay awake that night, the yellow sodium vapor glow from the streetlight casting shadows of tree branches on the wall. They curved and zagged in the illuminated square, and Rick tried to envision them as lines on a graph. How could they illustrate Fermat’s last theorem?

Fermat’s was one of the oldest—and simplest—unproved theorems in mathematics. Pythagoras had proved that the sides of a right triangle added up by the relationship A² + B² = C², and it was easy to show that there were an infinite number of integer solutions to the equation, but no one had been able to find an integer situation in which A³ + B³ = C³, or for any higher powers either. Nearly everyone believed that no such solutions existed, but nobody had been able to prove it.

Nobody except Pierre de Fermat. He had been more of a hobbyist than a mathematician, but he was good at it. He had made some big advances in probability theory, and had nearly invented calculus. And in the margin of Diophantus’s Arithmetics, beside the section discussing integer solutions to A² + B² = C², he had written his theorem, adding, “I have discovered a truly remarkable proof which this margin is too small to contain.”

Mathematicians had been trying for over three centuries to figure out what the Frenchman had had in mind, but so far without success. Andrew Wiles had come up with a proof in 1993, but it wasn’t elegant and it used semistable elliptic curves and hyperbolic planes—math that Fermat couldn’t have known about. Fermat’s proof had to be simpler, maybe even visual.

Rick tried to recall the insight he’d had in the car, but it was gone. Vanished as completely as Fermat’s own proof, which Fermat had never bothered to write down even though he’d had another twenty-eight years of life to do it after scribbling his tantalizing note in the margin.

Marsha came by his office the next day to borrow his stapler. “I’ve got to tack up a few spreadsheets to make it feel like home,” she said.

“I know what you mean,” Rick said, uncovering the stapler from the piles of notes and student papers on his desk. He wondered again where he could have met her before. It bothered him that he’d misplaced the memory of an entire romance, if that’s what it had been. Maybe it had been a single drunken revel at a convention somewhere.

Marsha took the stapler from him, but instead of leaving she sat down in the chair beside his desk and said, “So are you actually working on Fermat’s last theorem?”

Rick laughed. “I wasn’t until Vince brought it up, but since then I’ve had a couple thoughts I might pursue. I doubt if it’ll come to anything, though.”

She shrugged. “You never know.” There was a moment of uncomfortable silence, then, “You never told me what your déjà vu experience was last night.”

He felt himself blushing. “I, uh, well, I suddenly remembered that we’d met before.”

“Oh,” she said. “Have we?”

That settled that. She didn’t remember it either, so they must not have. Not that he’d have minded; she was a striking woman, and a genuine memory of a night with her would have been a great discovery to dredge out of the dark corners of his mind. He sighed. “I thought so, but then a moment later I wasn’t sure.”

“That’s the way with déjà vu.”

Rick thought back to the party. “Didn’t you say you’d felt something, too?”

“Yes, I did. For a second there, I thought I’d been an instructor here before.”

“Wow, Vince had a flash, too. What do you suppose the probability is of that happening to three people simultaneously?”

Marsha laughed. “Well, how often do you get déjà vu? Once a week, maybe?”

“Who knows?” Rick leaned back in his chair. “I suppose that’s a fair ballpark figure, though.”

“And the experience lasts for what, a minute at the most?”

“Sure.” He could see where she was going now, so he said, “One minute out of let’s see… fourteen-forty in a day, so just about exactly ten thousand minutes in a week. Jeez, for three simultaneous events, that’s one in ten thousand cubed”

“One in ten to the twelfth,” Marsha said. “One in a trillion. Holy cow.”

Rick whistled softly. “Mysterious forces are about.”

“Yeah, sure.” Marsha shook her head. “Now figure the odds of some mysterious coincidence out of the thousands of things that happen every day, and suddenly it doesn’t seem so surprising. It’d have to happen again before I’d believe there’s anything going on.”

Rick held up his left arm. “We’ll have to synchronize our watches, so we can compare our experiences. Or—” he paused, considering his next words carefully. “Or spend more time together.”

She tilted her head. “Oh? What did you have in mind?”

“Well,” Rick said, wiggling his watch. “It’s about lunch time. Have you tried the faculty dining hall yet?”

“No, I haven’t.”

Rick stood up and pulled his coat off the stand by the door. “By all means, then, let me show you where it is.”

The sidewalks were even icier than the streets, and since Marsha was wearing smooth-soled shoes Rick offered her his arm. That was his excuse, anyway.

“So tell me about your ideas for Fermat’s last theorem,” she asked him as they crossed the open quad from the math offices to the dining hall.

He told her about the sudden flash of inspiration he’d had on the drive home the previous night, how he’d thought of solving it by showing the relationships among A, B, and C visually, but how he’d lost the thread before he could write it down.

“Maybe that’s why Fermat never wrote it down either,” Marsha said, laughing. “Maybe he forgot it after he wrote his note in the margin.”

“God, that’d be terrible,” Rick said. “I just had a glimmer of hope, but imagine having the whole proof in your head and forgetting—” He stumbled, and Marsha had to help him stay on his feet, but he didn’t start walking again.

“What is it?” she asked.

“Quick, do you have something to write on?”

She patted her coat pockets. “No. Why?”

“Arrgh! Neither do I, but I’ve got it. I know how to solve it!”

“You do?”

“I—think so.” He turned half around, contemplating a sprint across the ice to his office, but he knew he’d never make it. Not before the memory faded. It was fading already. “Here,” he said, gripping Marsha’s arm fiercely. “Listen. In order to get an integer solution, you’ve got to have three integers whose squares or cubes or whatever are close enough together that two of them can be added together to make the third, right?”

“OK,” Marsha said, nodding.

“All right, we know it works for squares, so let’s look at cubes. Three cubed is twenty-seven, and four cubed is sixty-four, so their sum is ninety-one, which is less than five cubed. So we know that three and four don’t work. So now look at the general case, where you try it with any three consecutive integers. If you can show that no solutions exist for any of them, then you’ve proved the theorem.”

“What about three cubed plus, oh, seven cubed?” Marsha asked. “Or ninety-seven cubed. Wouldn’t you have to test every possible combination of integers?”

“No, no,” Rick said excitedly. “You just look at consecutive ones, because they’re the limiting case. You don’t test for three and ninety-seven, because you test for ninety-six and ninety-seven, and if the sum of their cubes is less than ninety-eight cubed, then you know the sum for three and ninety-seven is going to be less.”

“Sure,” Marsha said. “But how do you show that for all cases? There’s still an infinite number of consecutive integers.”

“Graphically,” Rick said immediately. “Yes, of course! Graph all the cubes of all the possible A’s and B’s and C’s, and—and—damn.”

“What?”

“I’ve lost the thread.”

“No, no, think. Graph all the possible sums of consecutive cubes, and…” she trailed off hopefully. “And…?”

Rick shook his head. “And I’ve lost it. It blinked out like a popped soap bubble.”

She tugged him into walking again. “Come on, let’s get back to your office and think this through. You had something; we’ll track it down.”

Rick let her lead him back while he tried desperately to remember his thought. He squeezed his eyes shut, opened them again, and saw three students standing beneath one of the bare trees to the side of the path. Were they—? Yes, they were the same grad students who’d arrived late at the party last night. Great; he’d drawn a crowd. Before the day was out, Vince would no doubt hear about his newest professor and Rick brainstorming arm-in-arm in the middle of the quad.

They tried all afternoon to reconstruct his train of thought, using his computer to graph every combination of squares, cubes, fourths and so on they could think of, but whatever spatial relationship he had visualized in that moment in the quad refused to materialize on the screen.

What did happen, though, was that Marsha edged her chair a little closer and a little closer to his over the course of the afternoon, until she might as well have been sitting in his lap. As Rick’s frustration with Fermat’s last theorem grew, he began to pay more attention to the woman by his side. A sudden rumble made them both look down at their stomachs, then Rick broke out laughing. “Guilty,” he said. “And I’m guilty of making you skip lunch, too. Would you let me take you to dinner?”

She rubbed her eyes and leaned back in her chair. “Sure. But this time bring a notebook and a pen.”

Over steak and shrimp in a quiet corner of the Bayview Inn, Marsha said, “You know, I forgot all about it in the excitement, but when you had your flash of inspiration there in the quad, I had another shot of déjà vu myself.”

“Impossible.”

“No it’s not. The odds for two of us are just a hundred million to one.”

“Hah. What did you remember?”

She got an impish grin and rolled her eyes toward the ceiling, but she finally said, “Well, I saw a whole chain of events leading up to my asking you up to my apartment for the night.”

“You’re kidding.”

Marsha toyed with a shrimp on her plate. “I usually don’t kid about that sort of thing.”

“No, I didn’t mean it that way. I meant, That’s wild, because that’s the flash I got last night.”

“What?”

“Last night, at the party, when I said I’d just had a moment of déjà vu, that’s what I’d seen. You and me going to bed in an upstairs apartment. Yellow curtains on the windows, and flannel sheets with little sheep printed on them.”

“That’s my bedroom!” Marsha said. “Do you remember anything more about it?”

He shook his head. “Not about the room, but, um, you’ve got a couple of freckles just to the side of your left breast.”

She dropped her fork to the plate. “This isn’t really déjà vu we’re having, is it?”

“I don’t think so. It’s more like precognition or something.”

“Does that mean you’re going to solve Fermat’s last theorem?”

Rick shrugged. “Who cares? What I want to know is whether or not you’re going to invite me up to your apartment tonight.”

Her impish grin returned. “I don’t know. You haven’t kissed me yet. I never invite a man up to my apartment unless he’s a good kisser.”

“Well,” Rick said, leaning forward and puckering his lips, “I’m willing to take the test.”

Rick woke in the middle of the night with the answer fully-formed in his mind. He reached for the note pad he always kept on the nightstand, but his hand met something cold and hard that didn’t belong there, tipping it over with a crash. Oh, right. Different bedroom.

“What was that?” Marsha asked sleepily.

“Sorry,” Rick muttered. “I think I tipped over the lamp.”

“S’all right,” she said, snuggling up against him. “Worry about it in the morning.”

“But I—” need a note pad, he almost said, then he realized he didn’t after all. Whatever his thought had been, he’d lost it again.

He lay back down and stared at the ceiling. What the hell was going on? Marsha was right, this wasn’t just déjà vu. Something was happening to their minds.

Everything in the room had a silvery tint; the Moon had risen while they slept, and was now shining on the bed. A shadow drifted slowly across the foot of it, and Rick thought for a moment that it must be from a passing car’s headlights, but then he realized the angle was all wrong for that. It had to be something up in the sky.

He looked out the window and realized there was a third possibility: Someone was walking along the flat-topped roof of the next building over, and the Moon behind him threw his shadow directly into Marsha’s bedroom. It looked like the guy was carrying something on his shoulder, something that glinted in the moonlight.

A gun? No, it was too big for that, unless it was a bazooka. It looked more like a telescope. Oh, sure. The night was clear and the Moon was bright as could be; it was probably just an astronomy student out there.

Or a Peeping Tom. As Rick watched, the person on the roof set the telescope on a tripod and pointed it straight at Marsha’s window. When he bent down to look through it, Rick raised his head into the shaft of moonlight, middle finger extended.

The person with the telescope leaped back as if Rick had punched him in the nose. Rick slid out of bed and stepped to the window, just to let the guy know for sure he’d been spotted. He was about to open it and shout at the son of a bitch when the man took another step back, tripped over something, and disappeared.

“Oh shit,” Rick said.

Marsha sat up. “Now what?” she asked.

Rick swept his hand around in the darkness at the foot of the bed, looking for his clothes. “I think somebody just fell off the roof over there.”

“What?”

“Somebody was looking at us with a telescope. When I flipped him off, he backed up and dropped out of sight.”

“There was somebody looking at us?” Marsha asked.

“Don’t worry, he just got there.” Rick tugged on his pants and shirt, found his shoes and put them on without socks, and said, “Keep an eye on the roof. I’m going over there and look for a body on the ground.”

“Shouldn’t we call the police?” Marsha asked.

“Not until we know there’s a reason to,” Rick said. “It’d be better for both of us if we didn’t have our names in the paper.”

Marsha nodded. “Good point.”

Rick grabbed his coat on the way out. It was cold as hell stepping into the winter night straight from a warm bed, but he tugged the zipper up to his neck and stuck his hands in his pockets while he jogged across the courtyard, the snow squeaking beneath his shoes.

There was nobody on the ground on either side of the building. No tracks, either. The guy hadn’t gone over the edge, then, and he had to be still in the building. Rick tried the front door, but it was locked, so he went around to the side with the fire escape and leaped up, caught the counterbalanced ladder with one hand and pulled it down. Then, hoping nobody else had called the police either, he crept up the iron staircase three flights to the roof.

The telescope was still there, but Rick couldn’t see the guy who’d set it up. He approached the scope cautiously, looking for tracks in the snow, and sure enough, he found some. A semicircle had been trampled down behind the telescope, and four distinct prints led backward away from it where the peeper had retreated, but they didn’t go anywhere. They vanished into smooth snow, as if the guy had leaped into the sky.

Rick looked for the other end of the trail of footprints, the ones whoever it was had made when he approached, and he saw the same thing. Footprints suddenly appeared in the snow, leading a dozen feet or so up to the telescope.

Too strange. Rick turned once around, looking for anything he might have missed, but he was alone on the rooftop. Alone and cold. With a shrug, he waved at Marsha’s silvery form behind her window, then picked up the telescope and carried it back down the stairway. Let the Peeping Tom come after it if he wanted it back.

“It doesn’t have any lenses,” Marsha pointed out when they’d set it up in her living room. “Mirrors either.” Rick looked down the front of the tube and realized she was right, but when he aimed the telescope at the Moon and looked through the eyepiece, he saw a heavily-cratered surface that looked close enough to touch.

“Something’s certainly doing the job,” he said. He aimed it down to ward the one lit window in the building across from them and looked again. He could see into the room—a kitchen, it turned out—as clearly as if he had his face up against the glass, and now he heard a woman’s voice saying, “Come on, honey, get up or you’ll be late for work.”

“What did you say?” Marsha asked.

“I didn’t say anything,” Rick said, stepping aside so Marsha could have a look. “It sounded like it came from the telescope.”

“No way.” She peered into the eyepiece. Rick saw movement in the window, and Marsha said, “A woman in a blue bathrobe just walked past.”

The voice said, “I’ll cook you a couple of eggs. How do you want ’em?”

“Scrambled,” a male voice said.

“This is bizarre,” Marsha whispered, backing away from the telescope.

“It’s the NSA again,” Rick said disgustedly. “They’re the only people who could possibly have access to something like this.”

Marsha looked at him with wide eyes. Every mathematician lived in dread of the NSA. “What do we do?”

“Damned if I know,” Rick said. He examined the telescope again. “Nikon?” he muttered, reading the label on the side of the tube.

“Yeah, right.”

“No, really, that’s what it says.”

Marsha bent over to look, and that motion triggered another flash of déjà vu: Rick clearly remembered seeing her bend down to look at his proof of Fermat’s last theorem, the one where he graphed Aⁿ + (A + 1 )ⁿ and (A + 2)ⁿ, and the graphs proved that the two functions never intersected. But exactly what was being plotted against what?

Marsha gasped. “They’re going to break in,” she said.

“Who?”

“Them.” She pointed toward the roof across the way, where three figures had appeared. They were black silhouettes against the Moon, and each one turned around a couple of times, evidently looking for their ’scope. Rick pointed it toward them and heard a man’s voice say, “The doot must have come up and grabbed it.”

One of them turned and looked toward Marsha’s apartment. Rick reached out and switched off the light, but it was too late. The voice said, “Yada, there they are.”

“Now what?” another voice said.

“Rewind?” the third one, female, said.

“Nodo. I’ve already crossed myself twice,” the first voice said. “I do it again and I’ll spring a twonky a month long.”

“Bort that,” the woman said, “Let’s just go take it back.”

“But won’t that snark his discovery? We trang that up and the department’ll send us all back to the Cretaceous.”

“We’ve already tranged it. If he’s figured out how to work the scope, he’s probably—”

“Like it’s really tough to point it at something.”

“—Listening to us right now.”

“Oh skrot, you’re right. Let’s bounce before we snark it up any worse.” One by one all three figures disappeared again, and this time there was no doubt about them falling off the roof or hiding behind something. They simply vanished.

“That’s not the NSA,” Marsha whispered.

“No, I don’t think so,” Rick whispered back “But whoever they are, you were right; they’re coming here.”

“What can we do?”

“Have you got a gun?”

Marsha shook here head. “Just pepper gas,” she said, taking her purse from the table by the door and removing the leather-covered spray canister from inside it.

“Good,” Rick said. “Get ready to use it.” He went into the kitchen and grabbed the broom from the corner by the back door, busted off the bristled end by stomping on it, and swished the ragged broken end back and forth in the air. “This ought to do,” he said. “Now we stand back to back in the living room and wait for them to make their move.”

They took up their positions. “Don’t hesitate,” Rick said. “Spray ’em all the moment you see a target.”

“Right.” Marsha held out the pepper canister. “Shouldn’t we call the pol—”

But it was too late. With a flicker of light and a whuff of displaced air, the three figures from the roof materialized right beside them. Rick felt a moment of intense déjà vu; he could hardly see their black-clad shapes through the sudden visions of the elegant graph proving Fermat’s last theorem, but he swung around with the broomstick and flailed away at the shadowy figures beyond it.

Marsha spun and sprayed with the deadly accuracy of a city-bred woman, and all three intruders dropped to the floor, coughing and choking. A beam of blue light shot out from one of them and struck the wall beside Rick, and he leaped aside and whacked at the source of it with the stick. He felt wood strike flesh, and the light winked out. Holding his breath and blinking to regain his sight, he reached down and picked up a wide-barreled pistol-gripped flashlight and backed away again.

“Got any rope?” he asked.

It took fifteen minutes before any of their captives recovered well enough to speak. Rick and Marsha had tied them to kitchen chairs, and stood back to wait. In the bright overhead light, their stealth-black oversuits looked less threatening, almost silly.

“I recognize all three of them,” Rick told her. “They were at your welcoming party, and in the quad when we both got hit by that déjà vu. And I’d be willing to bet they were in the car that came at me from nowhere on the way home, too. Weren’t you?” He pointed the blue light gun at the most-recovered of the intruders.

“Hey, watch that,” the kid said, still sniffling from the pepper spray. “You pop me with that, you’ll twonk the whole connie.”

“You want to put that in English?” Rick asked.

“What part didn’t you snag?”

“Twonk.”

“Twonk? You’re tranged. I’m sure that’s a twencie. From way before your time, in fact. Means screwing up the time flow. Leaving a twonky, like a magnoscope on a rooftop.”

Marsha said, “I think he means an anachronism.”

“Straight! You twonk the connie—that’s continuum—and you cause all sorts of problems downstream.”

Rick leaned back against the kitchen counter. “Are you trying to tell us you’re time travelers?”

“Ain’t trying to tell you nada. You asked.”

Rick wouldn’t have believed a word of it if it hadn’t been for the telescope in the living room, and the way his head filled with future memories every time these guys showed up. Did time machines leave some kind of wake that caused déjà vu?

“What are you doing here?” he asked.

“What do you think? We’re here to dock you solving Fermat’s last theorem.”

“Dock?” Marsha asked.

“Doc-u-ment-a-rize,” the kid said. “It’s a field lab in historical vidding.”

Rick laughed. “You’re nuts. You’re tranged. I haven’t solved it. I wasn’t even interested in it until you guys showed up at the party last night and gave Vince the idea.”

“We didn’t give nobody zot.”

“Ah, but you did.” Rick tugged a paper towel off the roll beneath a cabinet and drew a quick XY axis on it with a pen from the cup by the phone, then he swept two curves upward from the origin. “That’s it, right?”

The kid’s eyes bugged. “Torrie, you get that?”

The girl coughed. “You skrottin’ me? I can hardly see.” Rick wondered how she could see anyway; her glasses—as lensless as the telescope—blinked from the frames with a dozen different colors, all aimed inward.

The kid said, “Skrot, that would’ve been maxie. The whole concept swished out in four lines on a napkin. Maybe we can get him to do it again. Will you?”

“Weren’t you listening?” Rick said. “I got the idea from you. Every time you show up, I see it in my head.”

There was a long silence, then the woman, Torrie, said, “Uh oh.”

“It’s leakage,” Vrank said. He was the talkative one, the narrator for the “dock.” He explained what Torrie and Shil, his camera and sound ops, were doing as they spread the guts of their time machines out on the kitchen table. Rick and Marsha had finally decided to let them go once they’d promised not to snark anything up.

The time machines looked like fanny packs until Shil unzipped their flaps and unfolded them into flat, floppy panels of circuitry. Vrank said, “There’s supposed to be suppressors in there to keep down the ripple effect when we chrono, but evidently one of ’em’s tranged. Sorry about that.”

“Sorry?” Marsha said. ‘You’ve put Rick through the wringer.”

“It’s worse than that,” Vrank said. “We’ve evidently twonked his whole life. You honestly weren’t even thinking about Fermat’s last theorem until last night?”

“Nope,” Rick said.

“What about the journal?” Torrie asked. “You wrote in your journal that you got a flash of inspiration at the reception, and again on the drive home, and you described how you worked on it for days after you met Marsha here, ’cause you wanted to impress her.”

“What journal?” Rick asked.

“Twonky,” Shil said, poking at a couple of bumps and a squiggle in the dark fabric. “The journal’s a—yada, sure enough, here’s the prob. It was yours, Vrank.”

“Skrot. Now what?”

“Can’t you go back and undo it?” Rick said, leaning over the table to look at the offending circuit. It looked like someone had doodled with a laundry marker on a piece of gray nylon. “Can’t you tell yourself to fix your time machine before you leave?”

Vrank shook his head. “For one thing, looping like that’s a good way to trang your brain, but even if I tried it, that’d just dox it up even worse. We came back here to dock your discovery, which means you do figure it out.”

“Or at least he gets the credit for it,” Torrie said.

Shil looked up from the time machine. He and Vrank and Torrie exchanged knowing glances. “It’d work,” Shil said.

“It would, wouldn’t it?” said Vrank.

“What?” Rick asked, backing warily away and getting ready with his broom handle. “What would work?”

“Going with it. Docking your discovery the way we originally intended to.”

“It’s not my discovery,” Rick said. “You guys planted it.”

“Yada? Then where did it come from?” Vrank asked. “We got it from you.”

There was a long, pregnant silence, then Rick pulled out a chair and sat down. “My head hurts,” he said.

“It’s not a proof,” he said the next morning. Torrie and Vrank and Shil and Marsha were gathered around Marsha’s computer, where Rick had displayed a graph of his brainstorm, but on the screen where the two curves were supposed to diverge and thus prove that no integer solutions existed, they crossed.

“That’s not right,” Vrank said. “You’re missing something.”

“Obviously.”

“Cheer up,” Torrie said. “You’ll get it soon. Your journal said it took you just a few days.”

“No pressure,” Marsha said, giggling.

“Yeah.” Rick stared at the screen for a few minutes, then said, “So Vrank, why don’t you jump forward and back again, give me another hint.”

Vrank laughed. “Sorry. Shil fixed the chrono, so it wouldn’t do any good.”

“Besides,” Torrie said, “we need to get some good shots of you thinking. It wouldn’t do to have you spring it right off, or people downtime would know for sure what happened.”

“Couldn’t have that,” Rick said sourly. He tapped at the keyboard.

changing the display from fourth to fifth power integers, but the parabolas still crossed.

“Actually,” Marsha said, “isn’t their intersection a counterexample? Maybe you can prove the theorem wrong. That would be just as big a deal.”

Rick thought about that, then said, “The intersection points aren’t necessarily rational numbers. Besides, Wiles proved the theorem true already. I’m just looking for a more elegant one. And trying to impress you, of course.” He put his arm around Marsha. She was the one good thing to come of all this.

A week later, he was getting used to having people from the future popping in and out of his office, his living room, even his bedroom. He drew the line at his bathroom, demanding at least one safe refuge, but that didn’t help him either. The theorem and his flawed proof of it haunted him no matter where he went, and that was what was driving him nearly crazy.

He called a halt to it the next time Vrank and his crew showed up in his office. “Look,” he told them, “it’s not going to work. The whole concept is bogus.”

“No, no,” Vrank assured him. “You’re close. I can tell.”

“Yeah, why don’t you?”

“What?”

“Why don’t you tell? Just give me the solution, I’ll write it down, you can finish your documentary, and I can get back to my life.”

Torrie frowned. “That wouldn’t make for good drama.”

“You want me to climb in a bathtub and shout ‘Eureka’? Fine. Write me a script. But I’m tired of trying to validate your mistake. I never wanted to work on this stupid theorem anyway.”

Shil looked up at the clock and grinned. Rick said, “What’s so funny?”

“Twenty minutes from now you won’t be saying that.”

“Oh?” Rick looked from Shil to Vrank to Torrie. All three had big grins on their faces. “What, this is the big moment?”

“Yada,” Torrie said. The telltales around her glasses winked red and yellow.

Rick looked back to the mess of papers on his desk. He’d given up on the graphical route, and was trying to come up with a completely new paradigm. “No way,” he said. “I’m completely at sea here. I couldn’t prove the Pythagorean theorem in twenty minutes.”

“Come on,” Torrie said, “try.”

Rick turned back to his desk, conscious of her glasses recording his every move, and tried to concentrate on the problem. What was he missing? He shuffled through the pages of graphs and the tables of logarithms, tapping his pencil on the tiny comer of bare desktop. Out of perversity, he picked his nose. If he concentrated, he could hear the clock humming, the second hand sweeping around and around.

When twenty minutes had come and gone, he looked up again and said, “I think it’s time for plan B.”

The proof from the future was no better than his own. Rick had called Marsha into his office to see it, but the two of them read the article in the futuristic plastic unibook with growing confusion. “It’s not a proof,” Rick said, using the recessed toggle on the side to flip back to the abstract on the first page. “It says it is, but it’s got the same problem I have with mine. Doesn’t anybody do peer review in the future?”

“That’s your article,” Vrank said, his voice worried. “I downloaded it straight out of the archive.”

“It may have my name on it,” Rick said, “but I’d never try to publish something with this basic a mistake in it.”

“Then it’s twonked,” Shil said. “Our little snark-up here must have rippled down the connie and changed it.”

“Oh, skrot,” Vrank said, grabbing the door frame for support. “Rick, you’ve got to solve it. If you don’t, we’re doxed bad.”

“Sorry,” Rick said. “I’ve given it my best shot.”

There was a moment of silence while everybody scratched their heads, then Marsha said, “Why don’t we go to the source?”

“We’re at the source,” Vrank said.

“No, I meant Fermat. Why don’t you go back to 1650 or whenever it was to get the proof from him?”

Vrank and his companions shuffled back and forth nervously, and finally Vrank said, “Well, uh, actually, we already did. He didn’t have a proof either.”

“You’re kidding.” Rick laughed. “He didn’t have a proof?”

Vrank shook his head. “We docked his life before we got to yours, just for a prologue, you know, but it turned out he was wrong when he wrote that note in the margin. He had a flash of inspiration, but it turned out to be—”

Rick’s laughter drowned him out. “Your time machine!” Rick said when he got his breath back. “The one with the bad filter. You did the same thing to him that you did to me.”

Vrank blushed. “Yeah, I guess so.”

“Then there never was a proof. Never was, and never will be. Except for Wiles’s semistable elliptic curves.”

“Yada.” Vrank sighed and turned to his companions. “We’re snarked, dooties.”

“What happens now?” Marsha asked.

“Nothing, at least from your point of view. We, on the other hand, get to bounce back downstream and see if anybody’s filed a complaint.”

“And if they did?”

“Then we have to pay a fine and make resti to anybody whose life got screwed up. Not to mention re-taking historical vid.”

“You don’t just go back and undo the mistake?”

Shil laughed. “Define ‘mistake.’ We came back to dock Rick solving the theorem, which he wouldn’t even have tried if we hadn’t. But we wouldn’t have come back here in the first place if he hadn’t done it already, so maybe we should trang the ripple suppressors in Vrank’s chrono again and bounce back and forth across the herenow until he twigs to it, hey? Which is the ‘real’ connie?”

“This is,” Rick said after a moment’s translation. “You’ve screwed with my head enough already; just leave well enough alone and go pay your fine.”

Vrank nodded. “That’s the way the judges usually figure it, too. Well, wish us luck.” He tapped his forehead in a mock salute, reached down to touch a control on his fanny pack/ time machine, and vanished. Torrie and Shil blinked out a moment later.

“Whew,” Marsha said, leaning back in the chair beside Rick’s. “I was afraid they were going to undo everything all the way back to me getting hired here, and I’d never have met you.”

Rick gave her a hug. “I’d have given them the proof before it came to that. I was just holding out to see if I could get them off our case entirely.”

“What?” Marsha stared at him with wide eyes. “You did figure it out?”

“That’s right.” Rick shuffled through the papers on his desk. “Hmm, that’s strange,” he said. “I had it here a minute ago.”

AUTHOR’S NOTE: In order to properly research this story, I, of course, had to find an elegant solution to Fermat’s last theorem. It is indeed a truly remarkable proof, which the margin of this publication is also too small to contain. Fortunately, the editor has graciously agreed to print it in a separate appendix, which can be found on page 179.