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.