FERMAT'S BEST THEOREM

by Janet Kagan

---for Isaac, a math story, and or Sndrew Wiles, whether or not. for the great grin


    Somewhere in the distance, Peter Kropotkin was saying, "...Divide this term by this term...."  Chalk slid and squeaked.

    Laurie Adamansky shifted in the hard wooden chair.  Abstracted, she laid aside her pencil to touch the snap of her breast pocket.  Secure.  Beneath her fingers folded paper crackled, the sound startlingly loud to her.  No-one else, of course, noticed.  Why would they?  It was the sort of April morning, bright through the chalk dust of a board-full of equations, that no-one noticed much of anything.

    She frowned slightly to herself and added, And why am I acting as if the damn thing might leap out and get away from me?  I came here with the full intention of handing the thing to Pete after class, didn't I?

    Peter went on with his talk.  Laurie picked up her pencil and tried to concentrate on what he was saying.  The staccato tack-tack of chalk came to an abrupt halt, but Peter's voice continued as he turned to face the class.  Still talking, he reached under the desk.

    I'm probably going to embarrass myself.  Just another crank solution, Laurie---you divided by zero somewhere and it's Spring and you can't see the mistake for looking.

    From beneath his desk, Peter produced a large cylindrical object.  Laurie noted, without really seeing it, that the object was a tomato juice can.

    Last week Peter had handed her just such a solution: "Here, kid, here's a puzzle for you.  Somebody claims to have solved your favorite problem.  I've been prodding it for a week---and I'm damned if I can find what's wrong with it."

    Peter, in real-time, still talking about the equation on the board, held up his left index finger as if to point to something.  Instead, he laid it extended on the edge of his desk.

    "Does there have to be something wrong with it?" she'd asked.

    "You tell me."  And he'd grinned at her and handed her the sheet of paper.  It had taken her two days, but she'd found what was wrong with it.

    With his right hand, Peter raised the can of tomato juice high above his head.  "So you see," he said in conclusion, "what we have here is a very simple but very elegant solution to the problem."

    And with that he slammed the can of tomato juice downward.  It struck his extended index finger with a BAM! that left his desk reverberating.  Chalk rattled across in front of him and fell to the floor, rattling to a halt only at the arch of Jimmy Rodriguez's sneaker.  Jimmy sat up as abruptly as Laurie had, shocked by the sound and sight; the chalk crunched beneath his foot.

    "Now," Peter went on, "let me show you what we can derive from this...."  He set aside the tomato juice can, picked up an eraser in his right hand, and went at the board again.

    The class, as one, shuffled and paid attention.  Jimmy shot Laurie a covert look, eyes widened.  Like Laurie, he was waiting for Peter to scream in pain.

    Laurie looked past him.  No, Peter was not going to scream in pain.  She looked at the tomato juice can: it was dented just where it had struck Peter's finger.  She looked back at Jimmy, shrugged and smiled.  Another of Peter's puzzles---why hadn't he smashed his finger to smithereens?

    Although Peter had obviously intended the stunt as an attention-getter, the puzzle served Laurie as a distraction until the end of the class.  When she rose and bent to gather up her books, the paper crackled in her pocket.

    Before she could lose her nerve again, she followed at Peter's heels, leaving the rest of the class to examine the dented can.  Without a word, she trailed him to his office and in.  Once inside, she couldn't bring herself to open the conversation.

    "Is something wrong?"

    At the words, she brought her attention at last to Peter, realizing with a start that her manner had caused him genuine concern.

    "Uh, no.  Nothing's wrong.  I mean, nothing's wrong and that's what wrong."

    Peter grinned at her.  "I hate spring," he said cheerfully.  "Spring is particularly hard on math students."  He settled into his chair and tipped so far back as to look precarious.  "Sit down.  Pretend it's winter."

    She couldn't sit.  Instead, she unsnapped her breast pocket and withdrew the paper, now well-crumpled.  She unfolded it and made a futile attempt to smooth it on the edge of his littered desk.  "Here," she said.  "I think I've got it."  As the words came out, she found her shoulders slumping.  "I found what was wrong with the one you were offered last year---you find what's wrong with mine."  She paused.  "I can't.  I've tried and tried, and it still looks right to me."

    Peter reached across his desk to retrieve the paper she'd laid on it.  A stack of papers slid to the floor.  Laurie jumped to lessen the disaster---it gave her a good excuse to ignore Peter's perusal of her work.

    "Aha!"  He peered over the edge of the desk at her.  He wasn't going to let her off the hook that easily.  "Yet another solution to Fermat's Last Theorem!  At least this one is the right length."

    That was a running gag.  Recently, all the proposed "solutions" had been computer-generated, running to pounds of printout paper and covering select cases of the theorem only.

    Fermat had claimed to have a solution easy enough (by implication) to recall but "which this margin is too small to contain."

    The crackpots too tended to proofs that covered six or seven pages, if you didn't count the photos and resumes they invariably included with their packages.

    "Off-hand, I can't see anything wrong with it," Peter said.

    Laurie straightened, so fast she almost lost the papers she'd been retrieving.  "Neither could I.  I've been over a thousand times now.  But, as you said, it's spring.  So there must be something wrong with it.  I just can't find it."  With exasperation, she slapped the stack of papers back onto Peter's desk.

    "Tell you what," Peter said.  "You go have a hot date---have several!---and let me worry about it.  I'll find your mistake, if it's there."

    "It had better be," Laurie said, grinning back at him.

    Peter's grin broadened.  "Why?"

    She hadn't stopped to consider that.  That certainly was the way she was behaving.  Not as if she'd solved something but as if she'd let some cat out of some bag.  It took her a minute to pin the feeling down.

    "Oh."  She could feel her grin turn sheepish.  "That's why."

    "Why?" he asked again.

    "That was the problem that lured me into mathematics as a major.  Something that looks so easy and yet has remained unsolved for so long.  Something you could solve with pencil and paper---none of this fourteen days' worth of computer time.  Something romantic...."

    "'Romantic'?  That's not a word I here applied to math very often!"

    "Romantic.  Peter, the man scribbles this in the margin of his book, then he goes off and gets himself killed in a duel, and the solution dies with him?  If that's not romantic, I don't know what is.  Helluva lot more romantic than what passes for a romance novel these days!"

    "I never thought of it that way."  Peter leaned back, quiet for a long moment, then he nodded.  "You're right.  I got hooked on Fermat's Last Theorem too.  'Romantic' never occurred to me as a reason but I believe you're right, in my case as well."

    He leaned forward.  "So why would you be happier to be wrong about this?"

    "For all the same reasons.  You got hooked by Fermat's Last Theorem; so did I.  If I've got it, what's to hook the next generation of kids?"

    "I see."

    Once begun, Laurie found she couldn't stop there.  "And think of all the things that have been found by people who were looking for the solution to that---the theory of ideals, for instance!"

    "Yes.  You're afraid that, if your solution is right, you'll be taking away a valuable...prod."

    "At least a valuable puzzle.  And puzzle-solving is what it's all about or you wouldn't be slamming tomato-juice cans on your finger.  How is your finger, by the way?"

    He held it up.  "Never felt better."

    "I thought as much.  Well, thanks, Peter.  I feel better.  I think I'll go promote myself that hot date you suggested.  Please let me know when you find where I've divided by zero, okay?"

#

    Half a dozen hot dates later, Laurie's springtime morale had perked up wonderfully.  None of them however distracted her from noticing that Peter had not yet found an error in her work.  She turned her attention to other things, most noticeably tomato-juice cans.  She smashed two pencils before she thought to check the dented can still sitting prominently on Peter's desk.  Aha!  The can Peter had used was a different brand---it did not have the reinforcing ridges that were used in the brand she'd bought.

    "I'm not telling," said Peter, who'd come in late.

    "I don't expect you to.  I expect to solve it myself."

    "Watch your fingers in the meantime."  He gave her hands a significant look.  "I see you have been."

    "'Kids!  Don't try this at home!'  I've been watching my fingers but I've seen an awful lot of splints this week.  Jimmy Rodriguez was sporting two today!"

    Peter chuckled.  "I know.  I should be ashamed of myself but I'm not.  I'm ashamed of him."

    "I'll take that as a clue.  It can't be solved by experimentation.  It has to be solved by theorem.  Speaking of theorems...."

    "I had to call in assistance.  You have a car, don't you?  An old friend of mine is coming in on the six o'clock train.  Werner Hochheimer.  Any chance you could pick him up for me?  If you do, I'll invite you to dinner with us."

    "Uh.  Peter?  That's like asking could I spare a day for Albert Einstein.  Of course I can pick him up, only---Peter, my car's an old clunker---I---"

    "Werner doesn't notice things like cars.  Clothes, either, but don't wear bluejeans---the maitre d' will.  It's one of those pretentious places."  He scribbled on a piece of paper.  "That's the train and the time; here's the address of the restaurant.  Meet you there."

    Laurie was too astonished to say anything.

    Peter said, "Don't worry.  The food's great.  I didn't pick the restaurant for its dress code."

    Just outside the door to Peter's office, Laurie had a severe attack of giggles.  Peter was easily as famous as Werner Hochheimer.  The only difference was that she knew Peter, so she thought of him as "Peter" not as some icon.  So, rationally, she could get to know Werner Hochheimer as "Werner --"  No, even her mind stubbornly refused to accept the idea of Laurie talking to Werner, which made her giggle again.

    Look at it logically, she tried to convince it.  Peter is a friend of---Professor Hochheimer's.  At least they're both in the same club.  Which bogged Laurie down all over again.  That club was "The Marginalia," and membership was by invitation only.  It consisted of seven of the most important mathematicians alive.  The curious thing about it was that it didn't consist of all the important mathematicians alive.  Laurie had never been able to determine the criteria for membership and had concluded it was a drinking club of sorts.  Which meant that Peter was a real friend of Professor Hochheimer's --

    Well, that logic hadn't worked.  Giggling, Laurie went off to her 3 o'clock class.  It was spring, after all.  Not even Peter expected her to be rational in the spring.  Spring created a sort of vacuum that sucked your mind out in a dozen different directions at once.

    Vacuum!  That was it!  If you bring the can down fast---at greater than 1g---then the can is moving faster than the liquid can fall....  She'd have just enough time after her class to change, test her theory, and pick up Professor Hochheimer.

#

    In magazine photos, Professor Hochheimer looked imposing.  In person, Laurie found him...well...cute.  He was a little rotund man with lively eyes.  Laurie wasn't more than five feet tall herself but she overtopped him.  The first thing he asked---after her name---was what sort of hijinks Peter was pulling in class these days.  By the time they'd gotten Professor Hochheimer's luggage into the car, she was "Laurie" and he was "Werner."

    "That many splinted fingers!  I marvel at your classmates---for their determination rather than brains, I'm sad to say.  I note that your hands show no such redecoration.  Are you not interested in Peter's puzzle?"

    Laurie had pulled the car to a stop at a red light.  She turned to him, raised an eyebrow and grinned.

    His face lit in a dimpled grin of its own.  "Aha!  You've solved it!  I see!  Then you'll give me a demonstration and we'll see if I can solve it as well, shall we?"

    "Okay."

    "Tell me," he said, as she started the car in motion once more, "have you heard the one about the engineer, the chemist, and the mathematician?"

    "Any number of them.  You tell me one, I'll tell you one."

    "An engineer, a chemist, and a mathematician all work for the same small firm.  Now, the manager of the firm has a very bad habit---he smokes cheap cigars.  Worse, he tosses his cigar butts in the wastepaper basket.  The result of this, as you might well guess, is the occasional wastepaper basket fire.

    "Well, the first time, the fire is discovered by the engineer---who tips over the wastepaper basket and stamps the fire out."

    Laurie giggled, already appreciative.  A couple of her hot dates had been engineers.

    "The second time, the fire is discovered by the chemist.  Now, the chemist quickly calculates the volume of the wastepaper basket, the amount of flammable material in it, and measures out the exact amount of water necessary...so that the very last drop of water extinguishes the very last spark...."

    Laurie turned the car into the restaurant parking lot and backed it into a space.  "Go on, Werner.  I can listen and park at the same time."

    "And the third fire is discovered by the mathematician...who looks down at the flames and says to himself, 'Hmm.  Wastepaper basket fire.  I can solve that.'  And he walks away."

    Laurie hadn't heard that one before.  She exploded into laughter.  "I spoke too soon.  I'm glad I wasn't still in the process of parking when you got to the end.  That one's a potential fender-bender!"

    Werner Hochheimer beamed at her.

    The two of them were still grinning as they walked into the restaurant.  It wasn't until halfway through dessert that Laurie started laughing all over again.

    Peter eyed Hochheimer and said, "Spring.  You remember what Spring does to grad students."

    "No," said Laurie.  "I just got the second part of the joke.  I asked a medievalist friend of mine once what the Latin motto of that club you two belong to meant.  She said it translated to 'I can solve that.'"

    "Caught," said Peter.

#

    The camaraderie of the previous night carried Laurie through her morning classes despite the rainy turn in the weather and the shoulder-aching heft of her tote bag---in with her books she also carried a tomato juice can, size large, ribless.

    She had promised to meet Peter and Werner in Peter's office before Peter's class.  Luckily, she had a free period to do it in.

    "Laurie, pull up a chair and sit down."  Peter looked unusually somber.  Seated behind Peter's desk, Werner Hochheimer was perusing a piece of rumpled paper.  He too seemed somber compared to the high spirits of the previous night.

    Frowning, Laurie moved a stack of papers off the third chair, drew it up, and sat.  "What's wrong, Peter?"

    Werner Hochheimer looked up from the paper, beamed at her.  "Nothing's wrong, Laurie.  Your solution is quite correct."

    Laurie let go her tote-bag.  It hit the floor with a tremendous thunk.  "The Fermat?  You mean my solution to Fermat's Last Theorem is correct?  It can't be!"

    "An odd choice of words," Werner Hochheimer said.  "Why, pray tell, 'it can't be'?"

    "Because that ends the puzzle.  Because---"  And before she knew it, she was telling Werner Hochheimer the same objections she'd raised to Peter.  How Fermat's Last Theorem had drawn her into the field, how the search for a solution had lead to such other fascinating developments.  "But mostly, I wonder what's left as a prize for the younger kids."

    Werner Hochheimer was nodding.  "I had the same concern.  So did Peter.  Now you must make the decision.  I assure you, your solution is correct."

    "I solved it."  The words came out flat...and then the realization grew and grew until the effect was headier than a dozen Springs all rolled into one afternoon.  "I solved it!"  She got to her feet and stood ten feet tall at least.

    From her elevated position, she looked wildly down at the two men.  "I feel like Alice in Wonderland," she said, "I'm surprised my head hasn't hit the ceiling."

    Peter and Werner were both smiling up at her, their expressions oddly expectant.  Puzzle, she thought, another of Peter's puzzles.  Oh, my!  It had taken Werner Hochheimer too short a time to determine that there were no errors in her solution.

    She sat down abruptly.  Putting her elbows on the edge of Peter's desk, she stared intently at Werner Hochheimer.  A small club...of seven mathematicians only...its motto "I can solve that!"

    "Oh, my," she said aloud.  "I'm the eighth to solve Fermat's Last Theorem."

    "No, Laurie," said Peter sharply.  "You're the first.  That was our agreement.  If you decide to publish, it's your name that goes in the math books.  You solved Fermat's Last Theorem."

    "And the Marginalia disband," she said.

    "This is true," said Werner Hochheimer.  "There would be no longer be a criterion for membership."

    "Or," Laurie felt the grin spread clear across her face, achingly broad, "you could make me a member of the club."

    "Indeed, we could."  Werner Hochheimer reached into the pocket of his jacket and brought out a small jeweler's box.  He handed it across the desk.

    Inside the box, Laurie found a small gold and enamel pin.  Written in a curve around its edge was the Latin motto of the Marginalia.  "'I can solve that!'" she said.  Then she looked up to meet Werner Hochheimer's eyes.  "I accept," she said.

    "Wait a minute, Laurie," said Peter.  "You must understand that this means you will not publish your solution.  You must understand that this means the next person to solve Fermat's Last Theorem will be the first person to solve it.  You must understand that, aside from your investiture in the Marginalia, you'll get no applause."

    "I understand, Peter," she said solemnly.  Then with mock outrage, she added, "You think I'm a one-shot?  I'll get my applause sooner or later---and my name in the math books, too."

    "Take some time.  Think about it before you decide."

    "I don't need to.  I've done my thinking about it.  I've spent the last few weeks more worried that you wouldn't find errors in my solution than that you would."  She took the insignia from its box and pulled the cap from the pin.  Handing the two pieces across the desk, she said, "Pin me, Werner?"

    Grinning, he came around the desk to do so.  Then he stood off to admire his handiwork.  "There.  We're pinned.  We'll do something a little more formal when we all get together at the next Association conference but, meanwhile, enjoy it."

    "Oh, I will!"  She couldn't help but reach up to touch the pin on her collar.  "Hey!  Are all eight solutions identical?"

    "Some are more elegant than others.  Don't worry, you'll see them all."

    A loud sound from somewhere beneath the papers on Peter's desk startled her.

    "Alarm clock," Peter said, fishing it out to turn it off.  "Class.  Want to sit in, Werner?"

    "Yes."  He winked at Laurie, who winked back out of pure good spirits.  "I haven't seen one of your Spring puzzles since that conference in Buenos Aires."

    Through the hallways, Werner Hochheimer treated Laurie to a lively description of Peter's hijinx in Buenos Aires.  Laurie was still laughing when they reached the classroom.

    As always, Peter went first to the blackboard to erase the couplets Meijin Thomas always left behind.  Laurie settled Werner Hochheimer in her usual seat, motioned Jimmy Rodriguez over one and plopped her tote down to reserve herself a seat next to Werner.

    Instead of sitting, she fished out the tomato juice can and, cradling it in her arm, she walked to the front of the class.  Peter turned from the board just as she set the can down on the desk with an audible thunk.

    Werner Hochheimer watched, twinkling and patient.  Behind her, she could feel Peter's eyes as she balanced the huge can in her right hand, getting the balance just right.

    She laid her index finger on the table.  In front of her the class held its collective breath as she raised the tomato juice can high.

    She who hesitates, Laurie thought, gets a mashed finger.  And with that she slammed down the can of tomato juice as hard and as fast as she could.

    Blam!

    Everyone in the class gave a satisfying wince at the sight and a jump at the sound.

    Laurie righted the indented can and set it once more on the lab table.  She turned to Peter.  "I can solve that," she said, just loudly enough for the rest of the class to hear.  Then she took her seat next to Werner Hochheimer.

    He led the applause for her.

                                                   *end*


"Fermat's Best Theorem" 1995 by Janet Kagan. First published in Absolute Magnitude, Issue #3 Summer 1995, DNA Publications. Reprinted by permission of the author, Janet Kagan.
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