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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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