From: avjohn@iiasa.ac.at (Aviott  JOHN)
Subject: mathematical proof
Organization: IIASA, Laxenburg, Austria
X-Newsreader: TIN [version 1.1 PL6]
Date: Fri, 18 Dec 1992 06:58:33 GMT


MATHEMATICAL PROOF
by

Aviott JOHN

avjohn@iiasa.ac.at



The telegram from the U.S. National Academy of Sciences was brief. 
It raised much consternation in the offices of its counterpart in
distant Beijing.

The Secretary of the Chinese Academy raised both hands in horror
and sent for an assistant.  The bush-shirted assistant hurried into
the Secretary's office, ducking and bowing at the same time.

 'Chen Chi-shui, who is this man?  How do the Americans know
anything about him?'

Chen Chi-shui respectfully snatched the telegram from his superior
and avidly read its contents.  As he read, his jaw dropped
considerably and he made clucking disconcerted noises with half an
eye cocked on his chief, judging the suitability of his reactions and
adding intensity accordingly.

 'I do not know, respected sir.  I have never heard of this man.'

 'That is not surprising.  You know nothing about mathematics.  You
are a political appointee,' said Fang Lu, forgetting for the moment
that he was a political appointee himself, with a Ph.D. degree in the
interpretation of the Chairman's thoughts, a degree instituted at the
Xi'an Jiaotong University in the aftermath of the Cultural
Revolution.  Chen Chi-shui bowed his head and did not reply.

 'Find out,' Fang Lu softly instructed his assistant, using the tone
of voice he reserved for matters of the greatest urgency.  'Send
messages to the mathematics departments of all our Universities and
find out who is this man.'

Chen Chi-shui knew nothing about mathematics.  He was a
microbiologist by training and there was a ten year gap in his
knowledge caused by "the troubles," the current euphemism for that
turbulent period when the Red Guards kicked him out of his University
post and arbitrarily sentenced him to hard labor in a
coal mine in Shanxi Province.  The present administrative post
in the Chinese Academy had been arranged by his uncle, who was Party
secretary for the Shan Qian prefecture north of Beijing.

Chen ran from the Secretary's air-conditioned office, ignoring the
elevator bank and hurrying down the stairs to the telex office on the
ground floor.  The elevators were mostly switched off to conserve
electricity and were turned on only during the visits of VIPs,
members of the Politburo or Academicians from abroad.

 'Urgent, urgent,' he said, waving the telegram in the telex
operator's face.  'This message must go out immediately to all our
Universities.'

 'Write out the message on this form and we will do our utmost,'
said the telex operator with a smug look on her face.  She knew her
power in this vital position and used it unhesitatingly.

 'This is very urgent.  Fang Lu wants it out today.'

 'Fill out this form,' she repeated, but her attitude changed subtly
at the mention of the Secretary's name and she reached for the
battered directory of telex numbers that hung from a cord on her
desk.  This was probably going to cost her her lunch break, and she
had so looked forward to going window shopping for refrigerators that
day.

 'Do you know of a mathematician named Wang Zhao in your university
or anywhere else in China?  Has he written a famous paper on number
theory that was published in an American mathematical journal?  If
yes, inform Beijing immediately.  He is to come here at once.  All
his expenses will be paid.  Fang Lu has authorized the payment of air
fare or Soft Class train fare.'  The last sentence was a sweetener
that Chen had persuaded Fang Lu was necessary to underline the
urgency of the search and to ensure prompt response.  Many people
would respond in the hope of getting a plane ride.  Fang Lu had
grudgingly conceded the sense of this.

The rosy-cheeked telex operator set aside all thoughts of window
shopping with her sister-in-law at lunch and settled down to send the
message to dozens of Universities around the country, beginning with
the most famous.  Where the line was dead, she set aside the number
to try again later.

By the end of the week, eighty percent of the universities had
responded.  Five of them had Wang Zhao's in their mathematics
departments.  Of these, four were working in completely different
fields.  The fifth had specialized in number theory, but absolutely
denied having done any advanced work or published in any American
journal.  The Vice-Chancellor of the University concerned sent a
message confirming the man's claim.

It was a steamy afternoon in late summer.  The rains would soon
come to this part of the world.  Ho Win stood ankle deep in the muddy
water.  In his left hand he held a bundle of rice seedlings and
patiently, laboriously thrust the seedlings one by one into the
soft soil.  It was late in the evening when he reached the earth wall
that marked the edge of the small rice field.  He straightened up
with one hand pressed into the small of his back and wiped his damp
brow with the back of the other.  A trickle of muddy water ran down
his forearm and dripped off his elbow, leaving a trail of wet soil on
his skin.  The seedlings he had planted looked like emerald needles
emerging spikily from the muddy water.

A herd of water buffaloes plodded heavily by on the earth road,
looking for all the world like Party hacks returning after an
official lunch.  They were herded by a little boy riding on the neck
of the last one, urging on his charges with shrill cries rather like
an official responsible for ideological purity.  Ho Win grinned to
himself at the comparison, something he did so seldom that his
unlined face creased painfully.  He shouldered his ancient hoe and
walked slowly back to the University buildings a mile away, his eyes
turned inward to his thoughts, totally oblivious to the sharp stones
that stabbed at the soles of his feet through the cracked rubber
slippers he wore.  One of the rubber thongs plopped out of its slot
from time to time and he automatically stooped to slip it in place.

The main University building was a long, low, single-storied
structure of red brick, built by German missionaries before the war. 
The dun-colored tiles were fashioned from locally available clay
which was too coarse for the manufacture of the fine porcelain to
which China had given its name, but was ideally suited for the making
of roofing and bricks.

The light was fading fast as Ho Win shuffled to the rear of the
main building.  There were smaller buildings at the back, with clay-
brick walls and thatched roofs.  These were for the storage of sports
equipment and gardening tools.  He tossed the hoe into the smallest
of these.  The adjacent room was smaller than the tool shed, three
meters square and furnished with a kerosene lamp, a bed, chair and a
heap of clothes on the cement floor.  Ho Win picked up some clothes
from the pile on the floor and went out to wash and change by the
hand pump in the courtyard.  He tossed the damp field workers'
clothes on top of the pile and tiredly walked into the main building
through a rear entrance.  The building was deserted at this hour and
the corridors were almost totally dark, but he confidently made his
way to the library.  The library was locked, but he took the key from
its hiding place on the second ledge of the third window to the left
and entered the vestibule.  The air in the vestibule was moist and
smelt musty although it had been open only a few hours ago.

The book he reserved had been left for him on the librarian's desk. 
Switching on a table lamp, he began to read.  He had three hours time
to work and think before the lights went out at ten, a recently
instituted austerity measure.  As he read, Ho Win made notes on
scraps of paper.  Engrossed in his work, he did not see the shadowy
figure enter the library.

 'Ah, Professor Win.  How are you today?'

 Ho Win did not look up.  'Well, well,' he said impatiently and
waved the visitor away.

The Rector of the University was a kindly man but he did not
appreciate being ignored by a manual worker.

 'The Vice-President has given you permission to use the library,
but only provided I do not object.  Kindly sit up when I talk to
you.'

 Ho Win straightened abruptly.  'Yes, yes sir,' he said, instantly
respectful.  'What did you say?'

 'What are you working on?'

 'A difficult proof in number theory.'

 'Proof for what?'

 Ho Win sighed.  'Pierre de Fermat, a French mathematical
genius, wrote in the margin of a Greek text that he had
thought of a proof of his "last theorem," but the margin was
too small to contain it.  This has proved to be an irresistible
challenge to mathematicians the world over ever since.  The
problem remains unsolved and the question asked is, was this
alleged proof a joke on the mathematical world?  Many
mathematicians believe it cannot be proved.'

 'And you believe you can?'

 Ho Win shrugged modestly.  'I believe I have, but I need to
check the proof and that is what I'm doing now.'

 'What is this theorem anyway?'

 'If you want to know...' Ho Win sighed, wondering how he was
going to explain the concept to one who had majored in
political thought.  'As stated in Pythagoras' theorem, for
example, which I'm sure you'll know, A squared plus
B squared can equal C squared.  But can the sum
of two cubes be a cube, the sum of two numbers to the fourth
power be a fourth power, and so on.'

 'I see,' said the Rector, who obviously did not, stroking
his sallow cheeks.  'And you have a proof for this?'

 'I'm reasonably certain that I have proved that it cannot, but
there is a lot of work still needed.

 'And what will you do once this proof is completed?'

 'I will present it to the Vice-Chancellor and then he
can do what he likes with it,' Ho Win shrugged indifferently.

 'Actually I came to ask you a question related to your
mathematical work.'

 'What is it?' asked Ho Win worriedly, afraid that he might
lose his library privileges.

 'We have received an enquiry from the Academy in Beijing.  They
want to know the name of the Chinese mathematician who published a
paper on number theory in an American Journal.  They think the paper
is brilliant.  Do you have any idea who it could be?'

 'No.  Since the troubles, I have not had any contacts with other
mathematicians and what I wrote was only for my own pleasure.'

 'Hmm.  And you have no idea who it could be?'

 'If I see the paper, I might be able to guess.'

 'I will see if I can get you a copy of the paper.'

The Rector slipped out of the library as quietly as he appeared and
Ho Win bent to his work again.  Two days later Ho Win was summoned to
the Vice-President's office. 
Wan Wei-zi had white hair which made his unlined face look absurdly
youthful.  He had suffered many ups and down during the political
volte faces of the preceding decades and had learnt caution.  This
lesson was reflected in his eyes, but they were also compassionate.

 'Sit down.'  He motioned Ho Win to a chair opposite his modest
desk.  'Read this.  Do you know who wrote it?'

Ho Win took the mathematics journal and glanced at the article on
the open page. An algorithm for determining large twin primes,
by Wang Zhao.

'What is this?  This is my paper.  I wrote it some years ago.  Who
is Wang Zhao?'

 'Do you know how this paper reached the Americans?'

 'No, no.  I have no idea.' Ho Win's face wrinkled in remembrance
after a moment's reflection.  'Perhaps I do.  I was cleaning toilets
in Taching, in a correction camp on the oilfields.  One of my former
students slept in the same dormitory.  We often discussed mathematics
at night when we could not sleep for exhaustion after a day with the
driling rigs.  I got the idea for and wrote this paper while explaining
a concept to him.'

 'Where is this student now?  What is his name?'

 'I do not know where he is.  Perhpas he is in America.  His name
was Hu Yebang.'

The Vice-President was silent for so long that Ho Win rose to his
feet.  'Is there anything else I can
tell you?  I have to go back and
finish planting the rice.'

 'Of course.'  Wan Wei-zi nodded.  He had already forgotten Ho Win's
presence and was busy formulating a telegram to Beijing that would
put him in the most favorable light.

The air-conditioned office of the Secretary of the Beijing Academy
of Sciences was on the fifth floor.  Today he wore his best, a gray
three-piece pin-stripe suit and a blue polka-dotted tie, in honor of
the distinguished visitor.  The visitor was an American Nobel
laureate, an eminent mathematician who wore his laurels lightly. 
Professor Worth carried himself with the composure that comes with a
complete absence of humility, in marked contrast to the Secretary's
deference.  There was no servility in the Secretary's deference.  It
was rather a mask of civility, one which made as little impression on
the American as a fried egg on a teflon-coated pan.

Chen's office was a concrete cubicle on the third floor of the
faceless block building of the Academy of Sciences; an ugly, square
building like countless others that were springing up in the frenzy
of construction activity all over the capital.  The two year old
building was already beginning to crack, but it was still a mark of
privilege to work within it.  The elevators were turned on today in
Professor Worth's honor, so Chen Chi-shui used it to reach the
Secretary's office.  Chen wore his best blue shirt, gray wool
trousers and new black shoes and shivered a little in the coolness of
the air-conditioned office.

 'This is my assistant, Chen Chi-shui,' said Fang Lu.  'He is the
one who traced Wang Zhao for you.'

 'Where is he?  I'd like to meet him.  Our University is hosting
a conference this winter on Problems of Analytic Number
Theory and its Applications.  We would like Wang Zhao,
the author of this seminal paper on large primes, to be the
keynote speaker.

 'Why only Wang Zhao?'

 Professor Worth looked irritated by the question.  'Why
only Wang Zhao?  Because his paper is of extreme relevance
to concrete applications in high speed computing.  You
know, he is such a prize,
if some of the large American computer companies
knew he
was coming, they would pull out all stops and try to
seduce him with money to work for them.'

 'Our scientists are patriots,' said Fang Lu with a little
smile of confidence.

Fang Lu looked expectantly at Chen Chi-shui.  At that moment it
became awfully clear to Chen Chi-shui why he had been accorded the
privilege of meeting the distinguished visitor.  He writhed inwardly
with shame and at the loss of face.  To cover his discomfiture, he
uttered a shrill cackle of laughter.  Professor Worth's bushy
eyebrows rose over watery brown eyes as Chen Chi-shui stammered.

 'Wang Zhao's real name is Ho Win.  He has renounced mathematics and
unfortunately does not want to travel abroad.'

 'Perhaps I can speak to him and persuade him to change his mind.'

 'No, sorry, sorry.  He is very busy with other work and can come to
Beijing only after three weeks.'  Chen Chi-shui spoke the truth as he
knew it, for this was what Fang Lu had told him.

 'A pity.'  Professor Worth's forehead crinkled in annoyance.  'I
have to leave on Saturday.'  Chen Chi-shui emitted another yelp of
embarrassed laughter and Worth turned to Fang Lu.

 'Please do your best to persuade him to come.  Here is a copy of
the official letter of invitation to him.  Our University will pay
for his travel and look after all his expenses in the United States. 
In addition, he will receive a five thousand dollar honorarium if he
delivers the keynote address.  But we need to know by the end of this
month.'

 'I will do everything I can,' Fang Lu promised, 'but Ho Win is a
very busy man and, as my assistant just mentioned, he has a lot of
committments that will keep him busy for the next three weeks.' 
Which was the absolute truth.  The rice planting would last that
long.

 'I must congratulate you on this man.  He is a real genius.' 
Coming from Professor Worth it was arguably the highest accolade in
the mathematical world.

Fang Lu's chest expanded visibly at the American's words.  'Yes,
yes,' he nodded in a swell of national pride.  'In China
we have many talented people.'
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