aczel, amir d - fermat's last theorem_uc

8/10/2019 Aczel, Amir D - Fermat's Last Theorem_UC

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PRAISE FOR AMIR D. ACZEL'S

FERMAT'S LAST THEOREM

"This is a captivating volume. . . . The brilliant backdoor method

used by Mr. Wiles as he reached his solution, along with the debt he

owed to many other contemporary mathematicians, is graspable in

Mr. Aczel's lucid prose. !ually important is the sense o awe that Mr.

Aczel imparts or the hidden, mystical harmonies o numbers, and

or that sense o awe alone, his slender volume is well worth the

e#ort."

The New York Times

"$or more than three centuries, $ermat's %ast Theorem was the most

amous unsolved problem in mathematics, here's the story o how it

was solved. . . . An e&cellent short history o mathematics, viewed

through the lens o one o its great problems and achievements."

Kirkus Reviews

"This e&citing recreation o a landmark discovery reveals the great

e&tent to which modern mathematics is a collaborative enterprise. . .

. While avoiding technical details, Aczel maps the strange, beautiul

byways o modern mathematical thought in ways the layperson can

grasp."

Publishers Weekly

"(rie)y chronicles the history o the amous problem o the title,

which was recently solved by a mathematician named Andrew Wiles

ater he had devoted seven years to the task. . . . Aczel does a

superb *ob o creating in the nonmathematical reader the illusion o

comprehension."The New Yorker


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$3MAT'4 %A4T T563MUnlocking the Secret ojan ncient

!athematical Problem

By Amir D. Aczel

D e l t a


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A Delta Book

P!li"#e$ !y Dell

P!li"#i%& a

$ii"io% o(

Ba%tam Do!le$ay Dell P!li"#i%& )ro*+ I%c.

,-/ Broa$0ay

1e0 2ork+ 1e0 2ork ,//34

Co*yrit 5 ,664 !y Amir D Aczel

All rit" re"ere$. 1o *art o( t#i" !ook may !e re*ro$ce$ or tra%"mitte$ i%

a%y (orm or !y a%y mea%"+ electro%ic or mec#a%ical+ i%cl$i%& *#otoco*yi%&+

recor$i%&+ or !y a%y i%(ormatio% "tora&e a%$ retrieal "y"tem+ 0it#ot t#e

0ritte% *ermi""io% o( t#e P!li"#er+ e7ce*t 0#ere *ermitte$ !y la0. For

i%(ormatio% a$$re""8 For 9all" Eit 9i%$o0"+ 1e0 2ork+ 12+ ,//,,.

T#e tra$emark Delta: i" re&i"tere$ i% t#e ;.S. Pate%t a%$ Tra$emark O


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Pre"ace

7n 8une ,663+ my old riend Tom 4chulte was visiting me in (oston

rom 9aliornia. We were sitting at a sunny sidewalk cae on

:ewbury 4treet, tall, icy drinks in ront o us. Tom had *ust gotten

divorced, and he was ruminative. 5e hal turned toward me. "(y the

way," he said, "$ermat's %ast Theorem has *ust been proved." This

must be a new *oke, 7 thought, as Tom's attention was back on thesidewalk. Twenty years earlier, Tom and 7 were roommates, both o

us undergraduate students in mathematics at the ;niversity o

9aliornia at (erkeley. $ermat's %ast Theorem was something we

oten talked about. We also discussed unctions, and sets, and

number


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in mathematics, unaware o my own background in math, o#ered

me his help when we met at the 7nternational 5ouse where we both

lived. "7'm in pure mathematics," he said. "7 you ever have a math

problem you can't solve, eel ree to ask me." 5e was about to

leave, when 7 said ";rn, yes. There is something you can help me

with . . . " 5e turned back, "Why, sure, let me see what it is." 7

pulled open a napkinwe were at the dining room. 7 slowly wrote on

it?

#n$ yn% znhas no whole number solution when n is greater

than @.

"7've been trying to prove this one since last night," 7 said,handing him the napkin. 7 could see the blood draining rom his

ace. "$ermat's %ast Theorem," he grunted. "es," 7 said, "ou're in

pure math. 9ould you help me>" 7 never saw that person rom up

close again.

"7 am serious," Tom said,


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the accolades belong to others as much? Ben 3ibet, (arry Mazur,

Coro 4himura, utaka Taniyama, Cerhard $rey and others. This book

tells the entire story, including what happened behind the scenes

and out o the view o the media's cameras and )oodlights. 7t is

also a story o deception, intrigue, and betrayal.

7:


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&Perha's ( coul) best )escribe my e#'erience o" )oing

mathematics in terms oj entering a )ark mansion* You go into the

+rst room an) it,s )ark- com'letely )ark* You stumble aroun)-

bum'ing into the "urniture* .ra)ually- you learn where each 'iece

o" "urniture is* n) +nally- a"ter si# months or so- you +n) the light

switch an) tum it on* Su))enly- it,s all illuminate) an) you can see

e#actly where you were* Then you enter the ne#t )ark room * * * &

This is how Pro"essor n)rew Wiles )escribe) bis seven/year

0uest "or the mathematicians, 1oly .rail*


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8ust beore dawn on 8une @ 3+ , 66 3+ +roessor 8ohn 9onway

approached the darkened mathematics building on the +rinceton

;niversity campus. 5e unlocked the ront door and !uickly walked

up to his o=ce. $or weeks preceding his colleague Andrew Wiles'

departure or ngland, persistent but unspeci


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o thinning light hair. (orn in 9ambridge, his return was a very

special kind o homecomingit was the realization o a childhood

dream. 7n pursuit o this dream, Andrew Wiles had spent the last

seven years o his lie a virtual prisoner in his own attic. (ut he

hoped that soon the sacri


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comments that help them improve the papers beore they are

published. (ut Wiles didn't hand out preprints and didn't discuss his

work. The title o Wiles' talks was "Modular $orms, lliptic 9urves,and Calois 3epresentations," but the name gave no hint where the

lectures would lead, and even e&perts in his


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4himura0Taniyama 9on*ecture. Then suddenly he added one


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an "amateur" since he had a day *ob as a *urist, the leading histo0

rian o mathematics . T. (ell, writing in the early part o the

twentieth century, aptly called $ermat the "+rince o Amateurs."

(ell believed $ermat to have achieved more important mathe0

matical results than most "proessional" mathematicians o his day.

(ell argued that $ermat was the most proli


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Pierre $e Fermat'" La"t T#eorem a" re*ro$ce$ i% a% e$itio% o( Dio*#a%t"'

rithmetica *!li"#e$ !y Fermat'" "o% Samel. T#e ori&i%al co*y o( Dio*#a%t"

0it# Fermat'" #a%$0ritte% %ote #a" %eer !ee% (o%$.


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mat, who was 4econd 9onsul in the town o (eaumont0de0%omagne,

and o 9laire de %ong, the daughter o a amily o parliamentary

*udges. The young $ermat was born in August, , 4/ , baptized

August @/ in (eaumont0de0%omagne2, and was raised by his

parents to be a magistrate. 5e went to school in Toulouse, and was

installed in the same city as 9ommissioner o 3e!uests at the age

o thirty. 5e married %ouise %ong, his mother's cousin, that same

year, ,43,. +ierre and %ouise had three sons and two daughters.

6ne o their sons, 9lement 4amuel, became his ather's scienti


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diversion rom his hard work, and since he had to limit his social

lie, mathematics probably o#ered a much0needed break. And the

ideas o calculus were ar rom $ermat's only achievement. $ermat

brought us number theory. An important element in number theory

is the concept o a prime number.

Prime Numbers


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The numbers two and three are prime numbers. The number our is

not prime because it is the product o two and two? @ & @ .

The number + nine is not 3 & 36+ and

ten is not @ & -,/. (ut eleven again is a prime number since

there are no integers other than eleven itsel and one2, which can

be multiplied together to give us , , . And we can continue this

way? ,@ is not prime, ,3 is, , is not, ,- is not, ,4 is not, ,? is

prime, and so on. There is no apparent structure here, such as

every ourth number is not a prime, or even any more complicated

pattern. The concept has mysti


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All o $ermat's other theorems were either proved or disproved by

the early , >/ /" . This seemingly simple statement remained

unsettled, and thereore was given the name "$ermat's %ast The0

orem." Was it indeed true> ven in our own century, computers

were stymied in attempts to veriy that the theorem was true.

9omputers could veriy the theorem or very large numbers, but

they couldn't help or all numbers. The theorem could be tried on

billions o numbers, and there still would be in


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close riend o Andrew Wiles, watched him agonize every day over

the proo he had told the entire world he possessed only two

months earlier in 9ambridge. "7t's as i Andrew was trying to lay an

over0sized carpet on the )oor o a room," 4arnak e&plained. "5e'd

pull it out, and the carpet would


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roots o this theorem are grounded in the everyday lie o the

people o Mesopotamia o @/// (.9. .

The era rom @ // / (.9. to 4 // (.9. in the Mesopotamian

valley is considered the (abylonian ra. This time saw remarkable

cultural developments, including writing, the use o the wheel, and

metal works. A system o canals was used or irrigating large tracts

o land between the two rivers. As civilization )ourished in the

ertile valley o (abylon, the ancient people who inhabited these

planes learned to trade and to build prosperous cities such as

(abylon and %lr where Abraham was born2. ven earlier, by the

end o the ourth millennium (.9., a primitive orm o writing had

already developed in both the Mesopotamian and the :ile river

valleys. 7n Mesopotamia, clay was abundant and wedge0shaped

marks were impressed with a stylus on sot clay tablets. These

tablets were then baked in ovens or let to harden in the sun. This

orm o writing is called cuneiorm, a word derived rom the %atin

word cuneus- meaning wedge. The cuneiorm constitutes the


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Wealth (s a S;uare) 0uantity

A sophisticated number system was developed using base si&ty and

(abylonian engineers and builders were able to compute the

!uantities re!uired in their everyday proessional lives. 4!uares o

numbers appear naturally in lie, although it doesn't seem so at


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F E R M A T ' S L A S T T H E O R E M

&Plim'ton 7


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The (abylonians' approach was not to develop a general theory

or solving such problems, but rather to provide tables listing triples

o numbers andapparentlyto teach pupils how to read and use

these tables.

n ncient Society o" Number/Worshi''ers Sworn to Secrecy

+ythagoras was born on the Creek island o 4amos around 1L/ (.C.

5e travelled e&tensively throughout the ancient world and visited

(abylon, gypt, and possibly even 7ndia. 7n his travels, especially in

(abylon, +ythagoras came in contact with mathematicians and likely

became aware o their studies o numbers now named ater him

the +ythagorean triples,


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which (abylonian scientists and mathematicians had known about

(or over ,- // years. +ythagoras came in contact with the

builders o magni


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A M I R D . A C Z E L

&Number (s

3verything&

7n the barren, stark surroundings o the tip o 7taly, +ythagoras

ounded a secret society dedicated to the study o numbers. The

society, whose members became known collectively as the

+ythagoreans, is believed to have developed a substantial body o

mathematical knowledgeall in complete secrecy. The

+ythagoreans are believed to have ollowed a philosophy sum0

marized by their motto that "number is everything." They wor0

shipped numbers and believed them to have magical properties. An

ob*ect o interest to them was a "perect" number. 6ne o the

de + since

the numbers that can divide @> without remainder2 are ,+ @+ +

?+ and , + and we note that also? , G @ G G ? G ,

@>.

The +ythagoreans ollowed an ascetic liestyle, and were strict

vegetarians. (ut they did not eat beans, thinking they resembled

testicles. Their preoccupations with number were very much in the

spirit o a religion, and their strict vegetarianism originated in

religious belies. While no documents survive dating to the time o

+ythagoras, there is a large body o later literature about the

master and his ollowers, and +ythagoras himsel is considered one

o the greatest mathematicians o


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anti!uity. To him is attributed the discovery o the +ythagorean

Theorem concerning the s!uares o the sides o a right triangle,

which has strong bearing on +ythagorean triples and, ultimately, on

$ermat's %ast Theorem two thousand years later.

The S;uare oj the 1y'otenuse (s 3;ual to the

Sum oj the S;uares oj the =ther Two Si)es* * *

The theorem itsel originated in (abylon, since the (abylonians

clearly understood "+ythagorean" triples. The +ythagoreans,

however, are credited with setting the problem in geometric


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terms, and thus generalizing them away rom strictly the natural

numbers positive integers without zero2. The +ythagorean

Theorem says that the s!uare o the hypotenuse o a right0triangle

is e!ual to the sum o the s!uares o the two remaining sides o the

triangle, as shown above.

When the hypotenuse is an integer such as -+ whose s!uare is

@- + the general +ythagorean solution in terms o the sum o two

s!uares will be the integers our whose s!uare is si&teen2 and

three whose s!uare is nine2. 4o the +ythagorean Theorem, when

applied to integers whole numbers such as ,+ @+ 3+ . . .

gives us the +ythagorean triples which were known a millennium

earlier in (abylon.


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7ncidentally, the +ythagoreans also knew that s!uared numbers

are sums o se!uences o odd numbers. $or e&ample, , G

3 + = 6 , G 3 G - + = ,4 , G 3 G - G ?+ and so on. This

property they represented by a visual array o numbers in a s!uare

pattern. When the odd number o dots along two ad*acent sides is

added to the previous s!uare, a new s!uare is ormed?


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Whole Numbers- 8ractions- an) What 3lse>

(ut the +ythagoreans knew a lot more than whole numbers and

ractions numbers such as ,@+ ,3+ ->+ ,?,?46+ etc.2,

which were known in anti!uity both in (abylon and in gypt. The

+ythagoreans are the ones who discovered the irrational numbers

that is, numbers that cannot be written as ractions but have to be

written as unending, non0repeating decimals. 6ne such e&ample is

the number pi 3 .,,-6@4- . . . 2, the ratio o the

circumerence o a circle to its diameter. The number pi is

unending,0 it would take orever to write it down completely since it

has in


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number was not an integer, nor even a raction, a ratio o two

integers. This was a number with an unending decimal represen0

tation which did not repeat itsel. As is the case with pi, to write

down the e&act number that is the s!uare root o two

,.,@,3-4@ . . . 2 would take orever, since there are in-?,@>-?,@>-?,@>-?,@>-? . . . + etc., which

one could describe without having to actually write down every

digit2. Any number with a repeating decimal representation here,

the se!uence >-?,@ repeats itsel over and over again in the

decimal part o this number2, is a rational number, that is, a

number that can also be written in the orm alb- that is, the ratio o

two integers. 7n this e&ample, the two integers are ,3 and ?. The

ratio ,3? is e!ual to ,.>-?,@>-?,@>-?,@>-?,@ . .

. , the pattern >-?,@ repeating itsel orever.

The discovery o the irrationality o the s!uare root o two

surprised and shocked these diligent number admirers. They swore

never to tell anyone outside their society. (ut word got out. And

legend has it that +ythagoras himsel killed by drowning the

member who divulged to the world the secret o the e&istence o

the strange, irrational numbers.

The numbers on the number line are o two distinct kinds?

rational and irrational. %ooked at together, they


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around every irrational number there are in @3 =, >6 , taunted and ridiculed 9antor or his

theories about how many rational and irrational numbers0 there are.

Bronecker is known or his statement, "Cod made the integers, and all

the rest is the work o man," meaning that he did not even believe that

the irrational numbers, such as the s!uare root o two, e&istedE This,

over two millennia ater the +ythagoreans.2 5is antagonism is blamed

or having prevented 9antor rom obtaining a proessorship at the

prestigious ;niversity o (erlin, and ultimately or 9antor's re!uent

nervous breakdowns and his ending up at an asylum or the mentally

in


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F E R M A T ' S L A S T T H E O R E M The

Pythagorean ?egacy

An important aspect o +ythagorean lie, with its dietary rules and

number worship and secret meetings and rituals, was the pursuit o

philosophical and mathematical studies as a moral basis. 7t is

believed that +ythagoras himsel coined the words'hiloso'hy@ love

o wisdom,0 and mathematics/- that which is learned. +ythagoras

transormed the science o mathematics into a liberal orm o

education.

+ythagoras died around 1// (.9. and let no written records o

his work. 5is center at 9rotona was destroyed when a rival political

group, the 4ybaritics, surprised the members and murdered most o

them. The rest dispersed about the Creek world around the

Mediterranean, carrying with them their philosophy and number

mysticism. Among those who learned the philosophy o

mathematics rom these reugees was +hilolaos o Tarentum, who

studied at the new center the +ythagoreans established in that city.

+hilolaos is the


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were mystical. A diagonal point divides a diagonal into two une!ual

parts. The ratio o the entire diagonal to the larger segment is

e&actly the same as the ratio o the larger segment to the smaller

one. This same ratio e&ists in all smaller and smaller diagonals. This

ratio is called the Colden 4ection. 7t is an irrational number e!ual to

,.4,> . . . 7 you divide , by this number, you get e&actly the

decimal part without the , . That is, you get /.4,>. . . . As we

will see later, the Colden 4ection appears in natural phenomena as

well as proportions that the human eye perceives as beautiul. 7t

appears as the limit o the ratio o the amous $ibonacci numbers

we will soon encounter.

ou can .. . and / .4 ,> .. . inter0

changeably, once you have done the repetitive set o operations


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enough times. This is the Colden 4ection. 7t is e!ual to the s!uare

root o


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The +ythagoreans considered the tetraktys holy and even swore

oaths by it. 7ncidentally, according to Aristotle as well as 6vid and

other classical writers, the number ten was chosen as the base or

the number system because people have ten


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If the river carried away any portion of a man' !ot" the #in$ ent peron to

e%amine and determine &y mearement the e%act e%tent of the !o( From

thi practice" I thin#" $eometry firt came to &e #nown in E$ypt" whence it

paed into )reece(

Ceometry is the study o shapes and


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theorem is a rigorous *usti


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em limit arguments are not di#erent rom udo&us' "e&haustion"

method.

(ut the most brilliant mathematician o anti!uity wasundoubtedly Archimedes @>?=@,@ (.9.2, who lived in the city o

4yracuse on the island o 4icily. Archimedes was the son o the

astronomer +heidias, and was related to 5ieron 77, the king o

4yracuse. %ike udo&us, Archimedes developed methods or


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his mathematician riend to


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Archimedes' avorite theorem had to do with the sphere inside

the cylinder and he wrote the theorem in a book called The !etho)*

As with most ancient te&ts, it was assumed lost. 7n ,6/4 + the

Fanish scholar 8. %. 5eiberg heard that in 9onstantinople there was

a aded parchment manuscript with writings o a mathematical

nature. 5e traveled to 9onstantinople and ound the manuscript,

consisting o ,>- leaves o parchment. 4cienti


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7t is not certain when Fiophantus lived. We can date his period

based only on two interesting acts. $irst, he !uotes in his writings

5ypsicles, who we know lived around ,-/ (.9. And second,

Fiophantus himsel is !uoted by Theon o Ale&andria. Theon's timeis dated well by the solar eclipse which occurred on 8une ,4+ A.F.

34. 4o Fiophantus certainly lived beore A.F. 34 but ater ,-/

(.9. 4cholars, somewhat arbitrarily, place him at about A.F. @-/.

Fiophantus wrote the rithmetica- which developed algebraic

concepts and gave rise to a certain type o e!uation. These are the

Fiophantine e!uations, used in mathematics today. 5e wrote


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called the 9rusades, the Arabs ruled a )ourishing empire rom the

Middle ast to the 7berian +eninsula. Among their great

achievements in medicine, astronomy, and the arts, the Arabs

developed algebra. 7n A.F. 43@+ the prophet Mohammed

established an 7slamic state centered at Mecca, which remains thereligious center o 7slam. 4hortly aterwards, his orces attacked the

(yzantine mpire, an o#ensive which continued ater Mohammed's

death in Medina that same year. Within a ew years, Famascus,

8erusalem, and much o Mesopotamia ell to the orces o 7slam, and

by A.F. 4 , so did Ale&andriathe mathematical center o the

world. (y A.F. Q1/, these wars as well as the ones among the

Moslems themselves subsided and the Arabs o Morocco and the

west were reconciled with the eastern Arabs centered in (aghdad.

(aghdad became a center o mathematics. The Arabs absorbed

mathematical ideas as well as discoveries in astronomy and other

sciences rom the inhabitants o the areas they overcame. 4cholarsrom 7ran, 4yria, and Ale&andria were called to (aghdad. Furing the

reign o the caliph Al Mamun in the early L//s, the rabian Nights

was written and many Creek works, including uclid's 3lements-

were translated into Arabic. The caliph established a 5ouse o

Wisdom in (aghdad, and one o its members was Mohammed 7bn

Musa Al0Bhowarizmi. %ike uclid, Al0Bhowarizmi was to become

world0renowned. (orrowing 5indu ideas and symbols or numerals,

as well as Mesopotamian concepts and uclid's geometrical

thought, Al0Bhowarizmi wrote books on arithmetic and algebra. The

word "algorithm" is derived rom Al0Bhowarizmi. And the word

"algebra" is derived rom the


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Bhowarizmi's most well0known book?l 4abr Wa,l !ucjabalah* 7t was

rom this book that urope was later to learn the branch o

mathematics called algebra. While algebraic ideas are in the root o

Fiophantus'ritbmetica- thel 4abr is more closely related to the

algebra o today. The book is concerned with straightorward

solutions o e!uations o


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ety o the time brought him into contact with Arab mathematical

ideas, as well as Creek and 3oman culture. When the emperor

$rederick 77 came to +isa, $ibonacci was introduced to the emperor's

court and became a member o the imperial entourage.

7n addition to ?iber 0ua)ratorum- $ibonacci is known or another

book he wrote at that time, ?iber baci* problem about

+ythagorean triangles in $ibonacci's book also appears in a

(yzantine manuscript o the eleventh century now in the 6ld +alace

library in 7stanbul. 7t could be a coincidence,0 on the other hand,

$ibonacci might have seen that same book in 9onstantinople during

his travels.

$ibonacci is best known or the se!uence o numbers named

ater him, the $ibonacci :umbers. These numbers originate in the

ollowing problem in the ?iber baci/*

How many pair of ra&&it wi!! &e prodced in a year" &e$innin$ with a

in$!e pair" if in every month each pair &ear a new pair which &ecome

prodctive from the econd month on-

The $ibonacci se!uence, which is derived rom this problem, is

one where each term ater the + ,3+ @,+ 3+ --+ >6+ ,+ . . .

This se!uence which is taken to continue beyond the ,@

months o the problem2 has une&pectedly signi+ >,3+ ,3@,+ @,3+ 3--+ -->6+ >6,+ etc.

:ote that these numbers get closer and closer to "r- 0 l@.


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This is the Colden 4ection. 7t can also be obtained using a calculator

by repeating the operation lOl G lOl G , . . . . as described earlier.

3ecall that the reciprocal lO&2 o the Colden 4ection gives the same

number, less , . The $ibonacci se!uence appears everywhere in

nature. %eaves on a branch grow at distances rom one another that

correspond to the $ibonacci se!uence. The $ibonacci numbers occur

in )owers. 7n most )owers, the number o petals is one o? 3+ -+ >+

, 3+ @ , + 3 + - -+ or > 6 . %ilies have three petals, buttercups


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The $ibonacci numbersappear in sun)owerstoo. The little )oretsthat become seeds inthe head o thesun)ower are arrangedin two sets o spirals?one winding in aclockwise direction andthe other counter0clockwise. The numbero spirals in theclockwise orientation isoten thirty0our and thecounter0clockwise


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on a branch.


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The rectangle has appealing proportions. The Colden 4ection

appears not only in nature but also in art as the classic ideal o

beauty. There is something divine about the se!uence, and in act

the $ibonacci 4ociety, which is active today, is headed by a priest

and centered at 4t. Mary's 9ollege in 9aliornia. The 4ociety is

dedicated to the pursuit o e&amples o the Colden 4ection and

$ibonacci :umbers in nature, art, and architecture, with the belie

that the ratio is a git o Cod to the world. As the ideal o beauty,

the Colden 4ection appears in such places as the Athenian

+arthenon. The ratio o the height o the +arthenon to its length is

the Colden 4ection.

The Parthenon, Athens, Greece.


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The great +yramid at Ciza, built many hundreds o years beore

the Creek +arthenon, has ratio o height o a ace to hal the side o

the base also in the Colden 4ection. The gyptian 3hind +apyrus

reers to a "sacred ratio." Ancient statues as well as renaissancepaintings display the proportions e!ual to the Colden 4ection, the

Fivine 3atio.

The Colden 4ection has been searched or as the ideal o beauty

beyond )owers or architecture. 7n a letter to the $ibonacci 4ociety

some years ago, a member described how someone looking to .

The 2ossistsMathematics entered Medieval urope through $ibonacci's works

and rom 4pain, then part o the Arab world, with the work o Al

Bhowarizmi. The main idea o algebra in those days was to solve

e!uations or an unknown !uantity. Today, we call the unknown

!uantity "&," and try to solve an e!uation or whatever value "&"

may have. An e&ample o the simplest e!uation is? & 0 - / .

5ere, we use simple math operations to


56/177

When these ideas were imported to urope, the Arabic shai was

translated into %atin. 7n %atin, "thing" is res, and in 7talian it is cosa*

4ince the


57/177

Tartaglia revealed his method, on the condition that 9ardano keep it

a secret rom the rest o the world. When 9ardano later learned the

same methods rom another cossist, 4cippi0one del $erro ,-4=

,-@4+ he immediately assumed that Tartaglia got his system

rom this person, and elt ree to reveal the secret. 9ardano then

published the methodology or solving cubic e!uations in his ,--

book rs !agna* Tartaglia elt betrayed and became urious with

9ardano. 7n his last years, he spent much o his time viliying his

ormer riend, and he succeeded in diminishing 9ardano's

reputation.

The cossists were considered mathematicians o a lower level

than the ancient Creeks. Their preoccupation with applied problems

in pursuit o


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obtained a copy o the Creekrithmetica o Fiophantus, translated

it and published it as 9io'hanti le#an)rini rithmeticorum ?ibri Se#

in +aris in ,4@,. 7t was a copy o this book that ound its way to

$ermat.

$ermat's theorem says that there are no possible +ythagorean

triples or anything beyond s!uares. There are no triples o numbers,

two adding up to the third, where the three numbers are perect

cubes o integers, or ourth0powers o integers,

S;uares- 2ubes- an) 1igher 9imensions

A theorem is a statement with a proo. $ermat claimed to have had a

"marvelous proo," but without seeing and validating the proo no

one could call his statement a theorem. A statement may be very

deep, very meaningul and important, but without the proo that it is

indeed true, it must be called a con*ecture, or sometimes a

hypothesis. 6nce a con*ecture is proven it then can be called a

theorem- or a lemma i it is a preliminary proven statement which

then leads to a more proound theorem. +roven results that ollow a

theorem are called corollaries. And $ermat had a number o such

statements. 6ne such statement was that the number @ @ A J , G, was

always a prime number. This con*ecture was not only not proven,

hence not a theorem, it was actually proven to be wrong*This was

done by the great 4wiss mathematician %eonhard uler ,?/?=

,?>3 in the ollowing century. 4o there was no reason to believe

that the "last theorem" was true. 7t could be true, or it could be

alse. To prove that $ermat's %ast Theorem was alse all some0


59/177

one would have to do is to


60/177

correct or the e&ponents 3+ + - + 4 + and ?. 7t would be a long

way to in


61/177

hard's parents to allow him to pursue mathematics, since he would

become a great mathematician. %eonhard, however, continued with

the theology in addition to mathematics, and religious eelings and

customs would be a part o his entire lie.

Mathematicial and scienti


62/177

this way he could *oin his two riends who had such e&cellent

positions doing nothing but their own research in 3ussia.

uler ound mathematics in whatever he studied, medicine

included. 4tudying ear physiology led him to a mathematical

analysis o the propagation o waves. At any rate, soon an invitation

came rom 4t. +etersburg, and in ,?@? he *oined his two riends.

5owever, on the death o +eter's wie 9atherine, there was chaos at

the academy since she had been the great supporter o research. 7n

the conusion, %eonhard uler slipped out o the medical section

and somehow got his name listed with the mathematical section,

where he would rightully belong. $or si& years he kept his head low

to prevent the detection o his move, and he avoided all social

interactions lest the deception be discovered. All through this

period he worked continuously, producing volumes o top0rated

mathematical work. 7n , ?3 3 he was promoted to the leading

mathematical position at the academy. Apparently uler was a

person who could work anywhere, and as his amily was growing,

he would oten do his mathematics while holding a baby in one

arm.

When Anna 7vanova, +eter the Creat's niece, became empress o

3ussia, a period o terror began and uler again hid himsel in his

work or ten years. Furing this period he was working on a di=cult

problem in astronomy or which a prize was o#ered in +aris. A

number o mathematicians re!uested several months' leave rom

the academy to work on the problem. uler solved it in three days.

(ut the concentrated e&ertion took its toll and he became blind in

his right eye.

uler moved to Cermany to be at the royal academy there, but

did not get along with the Cermans, who en*oyed long


63/177

philosophical discussions not to his taste. 9atherine the Creat o

3ussia invited uler back to the 4t. +etersburg academy and he was

more than happy to come back. At that time, the philosopher Fenis

Fiderot, an atheist, was visiting 9atherine's court. The empress

asked uler to argue with Fiderot about the e&istence o Cod.Fiderot, in the meantime, was told that the amous mathematician

had a proo o Cod's e&istence. uler approached Fiderot and said

gravely? "4ir, a $ bin % #- hence Cod e&ists,0 replyE" Fiderot, who

knew nothing about mathematics, gave up and immediately

returned to $rance.

Furing his second stay in 3ussia, uler went blind in his second

eye. 5e continued, however, to do mathematics with the help o his

sons, who did the writing or him. (lindness increased his mental

ability to do complicated calculations in his head. uler continued to

do mathematics or seventeen years and died while playing with his

grandson in ,?>3 . Much o the mathematical notation we usetoday is due to uler. This includes the use o the letter i or the

basic imaginary number, the s!uare root o = , . uler loved one

mathematical ormula, which he considered the most beautiul and

put it above the gates o the Academy. The ormula is?

eR G - N /

This ormula has , and /+ basic to our number system,0 it has the

three mathematical operations? addition, multiplication, and

e&ponentiation,0 and it has the two natural numbers pi and e,0 and

it has i" the basis or the imaginary numbers. 7t is also visually

appealing.


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The Seven :ri)ges oj Konigsberg

uler was such an incredible visionary in mathematics that his

pioneering work on imaginary numbers and what today is called

comple& analysis2, was not his only innovation. 5e did pioneering

work in a


65/177

uler asked whether or not it was possible to cross all seven

bridges without passing twice on any bridge. 7t is impossible. 6ther

problems, which were studied in modern times and were posed

because o interest in the seven bridges problem, are the various

map0coloring problems. A cartographer draws a map o the world.

7n this map, each country is colored di#erently, to distinguish it

rom its immediate neighbors. Any two countries or states that are

separated completely rom each other may be colored in the same

e&act color. The !uestion is, what is the minimum number o colors

re!uired so that no two states that touch each other are in the

same color> 6 course, this is a general problem, not bounded by

how the map o the world looks today. The !uestion is really, given

all possible con Civen boundaries between

states in the ormer ugoslavia or in the Middle ast, with very

unusual lines between political entities, this general problem

becomes relevant in applications.

Mathematically, this is a topological problem. 7n 6ctober ,>-@+

$rancis Cuthrie was coloring a map o ngland. 5e wondered what

would be the minimum number o colors to be used or the

counties. 7t occurred to him that the number should be our. 7n

, >? 6 a proo was given that the number was indeed our, but

later the proo was ound to be alse. Almost a century later, in

,6?4 + two mathematicians, 5aken and Appel, proved what had

become known as the $our 9olor Map problem. To this day,

however, their proo is considered controversial since it made use o

computer work, rather than pure mathematical logic.


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.auss- The .reat .erman .enius

An alleged error in uler's proo or n N 3 that is, or cubes2 was

corrected by 9arl $riedrich Causs ,???=,>--. While most o the

renowned mathematicians o this time were $rench, Causs,

undoubtedly the greatest mathematician o the timeand arguably

o all timeswas unmistakably Cerman. 7n act, he never let

Cermany, even or a visit. Causs was the grandson o a very poor

peasant, and the son o a laborer in (runswick. 5is ather was harsh

with him, but his mother protected and encouraged her son. oung

9arl was also taken care o by his uncle $riedrich, the brother o

Causs' mother Forothea. This uncle was wealthier than 9arl's par0

ents and made a reputation or himsel in the


67/177

-+/-/+ written on it. Apparently, Causs 66,//

,// 66 6> 6?..... 3 @ , /

5e noted that the sum o each column was ,//+ so there was

nothing long to add up. 4ince there were ,/, columns, the sum o

all the numbers was ,/, & ,//,/+,//. :ow, either o the two

rows had the sum he neededall the numbers rom , to ,//.

4ince he needed only one o the two rows, the answer was hal o

,/+,//+ or -+/-/. Pery simple, he thought. The teacher,

however, learned a lesson and never again assigned the youngCauss a math problem as punishment.

When he was 6>.

Causs' results in number theory, which were shared with the

mathematicians o his day by regular correspondence, were o great

importance in all attempts by mathematicians to prove


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$ermat's %ast Theorem. Many o these results were contained in a

book on number theory which Causs published in %atin in ,>/,+

when he was @. The book, 9ist"uisitiones rithmeticae, was

translated into $rench and published in +aris in ,>/? and received

much attention. 7t was recognized as the work o genius. Causs

dedicated it to his patron, the Fuke o (runswick.

Causs was a distinguished scholar o classical languages as well.

6n entering college he was already a master o %atin, and his

interest in philology precipitated a crisis in his career. 4hould he

ollow the study o languages or o mathematics> The turning point

was March 3/+ ,?64. $rom his diary, we know that on that day

the young Causs decided de Causs' riend 5. W. M. 6lbers

wrote him a letter rom (remen on March ?+ ,>,4+ in which he

told Causs that the +aris Academy o#ered a big prize or anyone

who would present a proo or a disproo o $ermat's %ast Theorem.

Causs surely could use the money, his riend suggested. At that

time, as he did throughout his mathematical career, Causs received


69/177

(ut Causs would not be tempted. +ossibly he knew how

deceptive $ermat's %ast Theorem really was. The great genius in

number theory may have been the only mathematician in urope torealize *ust how di=cult it would be to prove. Two weeks later,

Causs wrote to 6lbers his opinion o $ermat's %ast Theorem? "7 am

very much obliged or your news concerning the +aris prize. (ut 7

coness that $ermat's Theorem as an isolated proposition has very

little interest or me, because , could easily lay down a multitudeo such propositions, which one could neither prove nor dispose o."

7ronically, Causs made great contributions to the branch o

mathematics known as comple& analysisan area incorporating the

imaginary numbers worked on by uler. 7maginary numbers would

have a decisive role in twentieth0century understanding o the con0

te&t o $ermat's %ast Theorem.

(maginary Numbers

The comple& number


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o = , + denoted by i. They were put on their own number line,

perpendicular to the real number line. Together, these two a&es

give us the comple& plane. The comple& plane is shown below. 7t

has many surprising properties, such as rotation being

multiplication by i*

Mlti*licatio% !y i rotate" co%terclock0i"e.

The comple& plane is the smallest , , + decades ahead o his time, Causs was

studying the behavior o unctions on the comple& plane. 5e

discovered some amazing properties o these unctions, known as

analytic unctions. Causs ound that analytic unctions had special

smoothness, and they allowed


71/177

or particularly neat calculations. Analytic unctions preserved

angles between lines and arcs on the planean aspect that wouldbecome signi=,>4. When the theory reemerged without Causs' name

attached to it years later, other mathematicians were given credit

or work on the same analytic unctions Causs understood so well.

So'hie .ermain

6ne day Causs received a letter rom a certain "Monsieur %eblanc."%eblanc was ascinated with Causs' book, 9iscjuisitiones

rithmeticae- and sent Causs some new results in arithmetic theory.

Through the ensuing correspondence on mathematical matters,

Causs gained much respect or Mr. %eblanc and his work. This

appreciation did not diminish when Causs discovered that neither

was his correspondent's real name %eblanc, nor was the writer o

the letters a "Mr." The mathematician writing so elo!uently to

Causs was one o very ew women active in the proession at that

time, 4ophie Cermain ,??4=,>3,. 7n act, upon discovering the

deception, Causs wrote her?

Bt #o0 to $e"cri!e to yo my a$miratio% a%$

a"to%i"#me%t at "eei%& my e"teeme$

corre"*o%$e%t Mr. Le!la%c metamor*#o"e #im"el(

i%to t#i" ill"trio" *er"o%a&e 0#o &ie" "c# a

!rillia%t e7am*le o( 0#at , 0ol$ %$ $i< clt to

!eliee...


72/177

A letter rom Causs to 4ophie Cermain, written rom (runswick on

Causs' birthday, as stated in $rench at the end o his letter?"(ronsvic ce 3/ avril ,>/? *our de ma naissance."2

4ophie Cermain assumed a man's name to avoid the pre*udice

against women scientists prevalent in those days and to gain

Causs' serious attention. 4he was one o the most important

mathematicians to attempt a proo o $ermat's %ast Theorem, and

to make considerable headway on the problem. 4ophie Cermain's

Theorem, which gained her much recognition, states that i a

solution o $ermat's e!uation with K- e&isted, then all three

numbers had to be divisible by -. OThe theorem separated $ermat's

%ast Theorem into two cases? 9ase 7 or numbers not divisible by -+

and 9ase 77 or numbers that are. The theorem was generalized toother powers, and 4ophie Cermain gave a general theorem which

allowed or a proo o $ermat's %ast Theorem or all prime numbers,

n- less than ,// in 9ase 7. This was an important result, which

reduced the possible cases where $ermat's %ast Theorem might ail

or primes less than ,// to only 9ase 77.4ophie Cermain had to

drop the disguise she was using when Causs asked his riend

"%eblanc" or a avor. 7n ,>/?+ :apoleon was occupying Cermany.

The $rench were imposing war


73/177

one to intercede on his behal with the $rench Ceneral +er0nety in

5anover.5e wrote his riend, Monsieur %eblanc, to ask i %eblanc might

contact the $rench Ceneral on Causs' behal. When 4ophie Cermain

gladly complied, it became clear who she was. (ut Causs was

thrilled, as seen rom his letter, and their correspondence continued

and developed urther on many mathematical topics. ;nortunately,

the two never met. 4ophie Cermain died in +aris in ,>3,+ beore

the ;niversity o Cottingen could award her the honorary doctorate

which Causs had recommended she receive.

4ophie Cermain had many other achievements to her credit in

addition to her contributions to the solution o $ermat's %ast

Theorem. 4he was active in the mathematical theories o acousticsand elasticity, and other areas in applied and pure mathematics. 7n

number theory, she also proved theorems on which prime numbers

can lead to solvable e!uations.

The :laDing 2omet o" i > a

Causs did much important work in astronomy, determining the

orbits o planets. 6n August @@+ ,>,,+ he


74/177

retreated rom 3ussia. Causs was amused. 5e was not unhappy to

see the mperor deeated ater $rench orces had e&torted such

high amounts rom him and his countrymen.

The 9isci'le

The :orwegian mathematician :iels 5enrik Abel visited +aris in

6ctober o ,>@4. There, he tried to meet other mathematicians

+aris was at that time a mecca or mathematics. 6ne o the people

who impressed Abel the most was +eter Custav %e*eune Firichlet

,>/-=,>-6+ a +russian who was also visiting +aris and

gravitated toward the young :orwegian, at


75/177

the book o the seven seals? The gited Firichlet is known as the

person who broke the seven seals. Firichlet did more than anyone

else to e&plain and interpret the book o his great master to the rest

o the world.

(esides ampliying and e&plaining the 9is0uisitiones- as well as

proving $ermat's %ast Theorem or the power


76/177

ter's position. When Causs died in ,>--+ Firichlet let his

prestigious *ob in (erlin to accept the honor o replacing Causs at

Cottingen.

Na'oleon,s !athematicians

The mperor o the $rench loved mathematicians, even i he wasn't

one himsel. Two who were especially close to him were

CaspardMonge ,?4=,>,>+ and 8oseph $ourier ,?4>=,>3/.

7n ,?6>+ :apoleon took the two mathematicians with him to gypt,

to help him "civilize" that ancient country.

$ourier was born in Au&erre, $rance, on March @,+ ,?4>+ but at

the age o eight he was orphaned and was helped by the local

(ishop to enter the military school. ven at the age o twelve,

$ourier showed great promise and was writing sermons or church

dignitaries in +aris, which they passed o# as their own. The $rench

3evolution o ,?>6 saved the young $ourier rom lie as a priest.

7nstead he became a proessor o mathematics, and an enthusiastic

supporter o the revolution. When the revolution gave way to the

Terror, $ourier was repulsed by its brutality. 5e used his elo!uence,

developed over years o writing sermons or others, to preach

against the e&cesses. $ourier also used his great public speaking

skills in teaching mathematics at the best schools in +aris.

$ourier was interested in engineering, applied mathematics, and

physics. At the cole +olytechni!ue, he did serious research in these

areas, and many o his papers were presented to the Academy. 5is

reputation brought him to the attention o :apoleon, and in ,?6>

the mperor asked $ourier to accompany him, aboard his )agship,

along with the $rench )eet o


77/177

dred ships headed or gypt. $ourier was to be part o the %egion o

9ulture. The %egion's charge was to "endow the people o gypt

with all the bene


78/177

AMIR D. ACZEL

Perio)ic 8unctions

The clearest e&ample o a periodic unction is your watch. Minute by

minute, the large hand moves around a circle, and ater si&ty

minutes it comes back to the e&act same place where it started.

Then it continues and in e&actly si&ty minutes it again returns to the

same spot. 6 course the little hand will have changed positions as

the hours go by.2 The minutes hand on a watch is a periodic

unction. 7ts period is e&actly si&ty minutes. 7n a sense, the space o

all minutes o eternitythe set o in


79/177

$e%e$ a" t#e "i%e (%ctio%. T#i" i" t#e eleme%tary

tri&o%ometric (%ctio% tat at "c#ool. T#e co"i%e

(%ctio% i" t#e #orizo%tal mea"re o( t#e arm. Bot#

"i%e a%$ co"i%e are (%ctio%" o( t#e a%&le t#e arm

make" 0it# a #orizo%tal li%e t#ro t#e ce%ter o(

t#e 0#eel. T#i" i" "#o0% !elo0.

As the train moves orward, the vertical height o the arm traces

a wave0pattern as seen above. This pattern is periodic. 7ts period is

K/ degrees. $irst the arm's height is zero, then it goes up in a

wave0like ashion until it reaches one, then it decreases, then it

goes to zero again, then to negative values until one, then its

negative magnitude decreases to zero. And then the cycles starts

all over again.

What $ourier discovered was that most unctions can be

estimated to any degree o accuracy by the sum o many the0

oretically in


80/177


81/177

lea$ to t#e $i"coery o( atomor*#ic (%ctio%" a%$

mo$lar (orm"0#ic# al"o #a$ a crcial im*act o%

Fermat'" La"t T#eorem t#ro t#e early t0e%tiet#

ce%try 0ork o( a%ot#er Fre%c# mat#ematicia%+ He%ri

Poi%care.

?ame Proo"

At t#e Marc# ,+ ,>? meeti%& o( t#e Pari" Aca$emy+

t#e mat#ematicia% )a!riel Lame ,?6-=,>?/

a%%o%ce$ ery e7cite$ly t#at #e #a$ o!tai%e$ a

&e%eral *roo( o( Fermat'" La"t T#eorem. ;%til t#e%+o%ly "*ecic *o0er"+ n- 0ere attacke$+ a%$ a *roo(

#a$ !ee% &ie% o( t#e t#eorem i% t#e ca"e" % 3+ +

-+ ?. Lame "&&e"te$ t#at #e #a$ a &e%eral a**roac#

to t#e *ro!lem+ 0#ic# 0ol$ 0ork (or a%y *o0er n*

Lame'" met#o$ 0a" to (actor t#e le(t "i$e o( Fermat'"

eatio%+ 7-/%G y%+ i%to li%ear (actor" "i%& com*le7 %m!er".Lame t#e% 0e%t o% to "tate mo$e"tly t#at t#e &lory"#ol$ %ot all &o to #im+ "i%ce t#e met#o$ #e"&&e"te$ 0a" i%tro$ce$ to #im !y o"e*# Lioille,>/6=,>>@. Bt Lioille took t#e *o$im a(terLame a%$ !r"#e$ a"i$e a%y *rai"e. Lame #a$ %ot*roe$ Fermat'" La"t T#eorem+ #e ietly "tate$+!eca"e t#e (actorizatio% #e "&&e"te$ 0a" %ot%ie t#at i"+ t#ere 0ere ma%y 0ay" to carry ott#e (actorizatio%+ "o t#ere 0a" %o "oltio%. It 0a" a&alla%t attem*t+ o%e o( ma%y+ !t it $i$ %ot !ear(rit. Ho0eer+ t#e i$ea o( (actorizatio%+ t#at i"+!reaki%& $o0% t#e eatio% to a *ro$ct o( (actor"+0a" to !e trie$ a&ai%.

()eal Numbers

T#e *er"o% to a&ai% try (actorizatio% 0a" Er%"t

E$ar$ m=mer ,>,/=,>63t#e ma% 0#o &ot

clo"er t#a% a%yo%e el"e i%


82/177

#i" time to a &e%eral "oltio% o( Fermat'" *ro!lem.

mmer+ i% (act+ i%e%te$ a% e%tire t#eory i%

mat#ematic"+ t#e t#eory o( i$eal %m!er"+ i%

attem*ti%& to *roe Fermat'" La"t T#eorem.Bummer'" mot#er+ a 0i$o0 0#e% #er "o% 0a" o%ly

t#ree+ 0orke$ #ar$ to a""re #er !oy a &oo$

e$catio%. At t#e a&e o( eitee% #e e%tere$ t#e

;%ier"ity o( Halle+ i% )erma%y+ to "t$y t#eolo&y

a%$ *re*are #im"el( (or li(e i% t#e c#rc#. A%

i%"it(l *ro(e""or o( mat#ematic" 0it# e%t#"ia"m

(or al&e!ra a%$ %m!er t#eory &ot t#e yo%&

mmer i%tere"te$ i% t#e"e area"+ a%$ #e "oo%

a!a%$o%e$ t#eolo&y (or mat#ematic". I% #i" t#ir$

year a" a "t$e%t+ t#e yo%& mmer "ole$ a $i(=

clt *ro!lem i% mat#ematic" (or 0#ic# a *rize #a$!ee% oNere$. Follo0i%& t#i" "cce""+ #e 0a" a0ar$e$

a $octorate i% mat#ematic" at t#e a&e o( t0e%ty=o%e.

Bt mmer 0a" %a!le to %$ a %ier"ity

*o"itio% a%$ t#ere(ore #a$ to take a teac#i%& o! at

#i" ol$ &ym%a"im #i "c#ool. He remai%e$ a

"c#oolteac#er (or a%ot#er te% year". T#root t#i"

*erio$+ mmer $i$ mc# re"earc#+ 0#ic# #e

*!li"#e$ a%$ 0rote i% letter" to "eeral lea$i%&

mat#ematicia%" o( #i" time. Hi" (rie%$" realize$ #o0

"a$ it 0a" to #ae "c# a &i(te$ mat#ematicia% "*e%$

#i" li(e teac#i%& #i "c#ool mat#. 9it# t#e #el* o("ome emi%e%t mat#ematicia%"+ mmer 0a" &ie%

t#e *o"itio% o( *ro(e""or at t#e ;%ier"ity o( Bre"la.

A year later+ i% ,>--+ )a"" $ie$. Diric#let #a$ take%

)a""' *lace at )otti%&e%+ leai%& !e#i%$ #i" ol$ o!

at t#e *re"ti&io" ;%ier"ity o( Berli%. mmer 0a"

c#o"e% to re*lace Diric#let i% Berli%. He remai%e$ i%

t#at *o"itio% %til #i" retireme%t.

mmer 0orke$ o% a 0i$e ra%&e o( *ro!lem" i%

mat#emat=


83/177

ic"+ ra%&i%& (rom ery a!"tract to ery a**lie$ee%

a**licatio%" o( mat#ematic" i% 0ar(are. Bt #e ma$e

#i" %ame (or #i" e7te%"ie 0ork o% Fermat'" La"t

T#eorem. T#e Fre%c# mat#ematicia% A&"ti%=Loi"

Cac#y ,?>6=,>-?+ t#ot o% a %m!er o(

occa"io%" t#at #e #a$ (o%$ a &e%eral "oltio% to t#e

Fermat *ro!lem. Bt t#e re"tle""+ carele"" Cac#y

realize$ i% eery attem*t t#at t#e *ro!lem 0a" mc#

!i&&er t#a% #e #a$ a""me$. T#e el$" o( %m!er"

#e 0a" 0orki%& 0it# al0ay" (aile$ to #ae t#e

*ro*erty #e reire$. Cac#y le(t t#e *ro!lem a%$

0e%t to 0ork o% ot#er t#i%&".

mmer !ecame o!"e""e$ 0it# Fermat'" La"t

T#eorem+ a%$ #i" attem*t" at a "oltio% le$ #im

$o0% t#e "ame (tile track take% !y Cac#y. Bt

i%"tea$ o( &ii%& * #o*e 0#e% #e reco&%ize$ t#at

t#e %m!er el$" i%ole$ (aile$ to #ae "ome *ro*=

erty+ #e i%"tea$ invente) %e0 %m!er" t#at 0ol$ #ae

t#e *ro*erty #e %ee$e$. T#e"e %m!er" #e calle$

i$eal %m!er". mmer t#" $eelo*e$ a% e%tirely

%e0 t#eory (rom "cratc#+ 0#ic# #e "e$ i% #i"

attem*t" to *roe Fermat'" La"t T#eorem. At o%e

*oi%t+ mmer t#ot #e %ally #a$ a &e%eral

*roo(+ !t t#i" %(ort%ately (ell "#ort o( 0#at 0a"

%ee$e$.

1o%et#ele""+ mmer $i$ ac#iee treme%$o"

&ai%" i% #i" attack o% t#e Fermat *ro!lem. Hi" 0ork

0it# i$eal %m!er" e%a!le$ #im to *roe Fermat'"

La"t T#eorem (or a ery e7te%"ie cla"" o( *rime

%m!er" a" t#e e7*o%e%t n* T#"+ #e 0a" a!le to

*roe t#at Fermat'" La"t T#eorem 0a" tre (or a%

i%%ite %m!er o( e7*o%e%t"+ %amely t#o"e t#at are

$ii"i!le !y re&lar *rime". T#e irre&lar *rime

%m!er" el$e$ #im. T#e o%ly irre&lar *rime" le""

t#a% ,// are 3?+ -6+ a%$ 4?. mmer t#e% 0orke$

"e*arately o% t#e *ro!lem o( t#e"e


84/177

irre&lar *rime %m!er" a%$ 0a" ee%tally a!le to

*roe Fermat'" t#eorem (or t#e"e %m!er". By t#e

,>-/"+ "i%& m=mer'" i%cre$i!le !reakt#ro"+

Fermat'" La"t T#eorem 0a" k%o0% to !e tre (or all

e7*o%e%t" le"" t#a% n % ,//+ a" 0ell a" i%%itely

ma%y mlti*le" o( *rime %m!er" i% t#i" ra%&e. T#i"

0a" ite a% ac#ieeme%t+ ee% i( it 0a" %ot a

&e%eral *roo( a%$ "till le(t ot i%%itely ma%y

%m!er" (or 0#ic# it 0a" %ot k%o0% 0#et#er t#e

t#eorem #el$ tre.

I% ,>,4+ t#e Fre%c# Aca$emy o( Scie%ce" oNere$ a

*rize to a%yo%e 0#o 0ol$ *roe Fermat'" La"t

T#eorem. I% ,>-/ t#e Aca$emy a&ai% oNere$ a &ol$

me$al a%$ 3+/// (ra%c" to t#e mat#ematicia% 0#o

0ol$ *roe Fermat'" La"t T#eorem. I% ,>-4+ t#e

Aca$emy $eci$e$ to 0it#$ra0 t#e a0ar$+ "i%ce it

$i$%'t "eem like a "oltio% to Fermat'" *ro!lem 0a"

immi%e%t. I%"tea$+ t#e Aca$emy $eci$e$ to &ie t#e

a0ar$ to Er%"t E$ar$ mmer+ (or #i" !eati(l

re"earc#e" o% t#e com*le7 %m!er" com*o"e$ o(

root" o( %ity a%$ i%te&er". T#" mmer 0a"

a0ar$e$ a *rize (or 0#ic# #e %eer ee% a**lie$.

mmer co%ti%e$ #i" tirele"" eNort" o% Fermat'"

La"t T#eorem+ "to**i%& #i" re"earc# o%ly i% ,>?.

mmer al"o $i$ *io%eeri%& 0ork o% t#e &eometry o(

(or=$ime%"io%al "*ace. Some o( #i" re"lt" are

"e(l i% t#e el$ o( mo$er% *#y"ic" k%o0% a"

a%tm mec#a%ic". mmer $ie$ o( t#e Q i% #i"

eitie"+ i% ,>63.

mmer'" "cce"" 0it# i$eal %m!er" i" ee% more

#ily *rai"e$ !y mat#ematicia%" t#a% t#e actal

a$a%ce" #e ma$e i% t#e "oltio% o( Fermat'"

*ro!lem "i%& t#e"e %m!er". T#e (act t#at t#i"

%ota!le t#eory 0a" i%"*ire$ !y attem*t" to "ole Fer=

mat'" La"t T#eorem "#o0" #o0 %e0 t#eorie" ca% !e

$eelo*e$


85/177

!y attem*t" to "ole *articlar *ro!lem". I% (act+

mmer'" t#eory o( i$eal %m!er" le$ to 0#at are

%o0 k%o0% a" i$eal"+ 0#ic# #ae #a$ a% im*act o%t#e 0ork o( 9ile" a%$ ot#er mat#ematicia%" o%

Fermat'" t#eorem i% t#e t0e%tiet# ce%try.

nother PriDe

I% ,6/>+ t#e 9ol("ke#l Prize o( o%e #%$re$ t#o"a%$

mark" 0a" oNere$ i% )erma%y (or a%yo%e 0#o 0ol$

come * 0it# a &e%eral *roo( o( Fermat'" La"t

T#eorem. I% t#e r"t year o( t#e *rize+ 4@,

"oltio%" 0ere "!mitte$. All o( t#em 0ere (o%$ to

!e (al"e. I% t#e (ollo0i%& year"+ t#o"a%$" more

"oltio%" 0ere "!mitte$+ 0it# t#e "ame eNect. I%t#e ,6@/"+ )erma% #y*eri%Qatio% re$ce$ t#e real

ale o( t#e ,//+/// mark" to %ot#i%&. Bt (al"e

*roo(" o( Fermat'" La"t T#eorem co%ti%e$ to *or

i%.

.eometry Without 3ucli)

1e0 $eelo*me%t" "tarte$ taki%& *lace i%

mat#ematic" i% t#e %i%etee%t# ce%try. a%o" Bolyai

,>/@=,>4/ a%$ 1icola" Ia%oitc# Lo!ac#e"ky

,?63=,>-4+ a H%&aria% a%$ a R""ia% re"*ectiely+

c#a%&e$ t#e (ace o( &eometry. By $oi%& a0ay 0it#Ecli$'" a7iom t#at t0o *arallel li%e" %eer meet+

t#e"e t0o 0ere a!le i%$e*e%$e%tly to (ormlate a

&eometrical %ier"e 0#ic# mai%tai%e$ ma%y o(

Ecli$'" *ro*ertie" !t allo0e$ t0o *arallel li%e" to

meet at a *oi%t at i%%ity. A $iNere%t &eometry ca%

!e "ee%+ (or e7am*le+ o% a "*#ere "c# a" t#e &lo!e.

Locally+ t0o lo%&it$e li%e" are *arallel. Bt i%

reality+ a" t#ey are (ollo0e$ to t#e 1ort# Pole+ t#e

t0o li%e" meet t#ere. 1e0 &eome


86/177

trie" "ole ma%y *ro!lem" a%$ e7*lai% "itatio%"

0#ic#+ %til t#at time+ "eeme$ my"terio" a%$

0it#ot a "oltio%.

:eauty an) Trage)y

A!"tract al&e!ra+ a el$ $erie$ (rom t#e (amiliar

al&e!ra tat i% "c#ool a" a "y"tem (or "oli%&

eatio%"+ 0a" $eelo*e$ i% t#e %i%etee%t# ce%try.

I% t#i" area+ t#e !eati(l t#eory o( )aloi" "ta%$" ot.

Eari"te )aloi" 0a" !or% i% ,>,, i% t#e "mall

illa&e o( Bor&=la=Rei%e+ ot"i$e Pari". Hi" (at#er

0a" t#e mayor o( t#e to0% a%$ a "ta%c# re*!lica%.

2o%& Eari"te &re0 * o% t#e i$eal" o( $emocracy

a%$ (ree$om. ;%(ort%ately+ mo"t o( Fra%ce 0a" att#at time #ea$i%& i% t#e o**o"ite $irectio%. T#e

Fre%c# Reoltio% #a$ come a%$ &o%e+ a%$ "o #a$

1a*oleo%. Bt t#e $ream" o( (ree$om+ eality+ a%$

(rater%ity #a$ %ot yet !ee% ac#iee$. A%$ t#e

royali"t" 0ere e%oyi%& a come!ack i% Fra%ce+ 0it# a

Bor!o% o%ce a&ai% cro0%e$ i%& o( t#e Fre%c#%o0

to rle to&et#er 0it# t#e *eo*le'" re*re"e%tatie".


87/177

Eari"te'" li(e 0a" "tee*e$ i% t#e lo(ty i$eal" o( t#e

reoltio%. He 0a" a &oo$ i$eolo&e+ a%$ #e &ae

"ome moi%& "*eec#e" to t#e re*!lica%". A" a

mat#ematicia%+ o% t#e ot#er #a%$+ #e 0a" a &e%i"

o( %*arallele$ a!ility. A" a tee%a&er+ )aloi"

a!"or!e$ t#e e%tire t#eorie" o( al&e!ra a%$

eatio%" k%o0% to accom*li"#e$ mat#ematicia%" o(

#i" $ay+ a%$0#ile "till a "c#ool!oy$eelo*e$ #i"

o0% com*lete "y"tem+ k%o0% to$ay a" )aloi" T#eory.

;%(ort%ately+ #e 0a" %ot to e%oy a%y reco&%itio% i%

#i" tra&ically "#ort li(e. )aloi" "taye$ * %it" at #i"

!oar$i%& "c#ool+ 0#ile eeryo%e el"e "le*t+ a%$ 0rote

$o0% #i" t#eory. He "e%t it to t#e #ea$ o( t#e Fre%c#

Aca$emy o( Scie%cet#e mat#ematicia% Cac#y

#o*i%& t#at Cac#y 0ol$ #el* #im *!li"# t#e

t#eory. Bt Cac#y 0a" %ot o%ly ery !"y+ #e 0a"

al"o arro&a%t a%$ carele"". A%$ )aloi"' !rillia%t

ma%"cri*t e%$e$ * i% a tra"# ca%+ %rea$.

)aloi" trie$ a&ai%+ 0it# t#e "ame re"lt. I% t#e

mea%time+ #e (aile$ to *a"" t#e e%try e7am" to t#e

L7ole Polytec#%ie+ 0#ere mo"t o( t#e cele!rate$

mat#ematicia%" o( Fra%ce &ot t#eir trai%i%&. )aloi"

#a$ a #a!it o( al0ay" 0orki%& t#i%&" ot i% #i" #ea$.

He %eer took %ote" or 0rote t#i%&" $o0% %til #e

#a$ actal re"lt". T#i" met#o$ co%ce%trate$ o% i)eas

rat#er t#a% $etail. 2o%& Eari"te #a$ little *atie%ce

(or+ or i%tere"t i%+ t#e $etail". It 0a" t#e &reat i$ea+

t#e !eaty o( t#e lar&er t#eory+ t#at i%tere"te$ #im.

Co%"ee%tly+ )aloi" 0a" %ot at #i" !e"t 0#e% taki%&

a% e7ami%atio% i% (ro%t o( a !lack!oar$. A%$ t#i" i"

0#at ca"e$ #im to (ail t0ice i% #i" attem*t" to &ai%

e%tra%ce to t#e "c#ool o( #i" $ream". T0ice i% (ro%t

o( t#e !lack!oar$ #e $i$ %ot *er(orm 0ell 0riti%&

t#i%&" $o0%+ a%$ #e &ot irritate$ 0#e% a"ke$ (or

$etail" #e "t $i$%'t co%"i$er


88/177

im*orta%t. It 0a" a tra&e$y t#at a%i%cre$i!lyi%telli&e%t yo%& *er"o% 0ol$ !e

e"tio%e$ !y mc# le"" a!le e7ami%er" 0#o col$

%ot %$er"ta%$ #i" $ee* i$ea" a%$ took #i"

relcta%ce to &ie triial $etail" (or i&%ora%ce. 9#e%

#e realize$ #e 0a" &oi%& to (ail t#e "eco%$ a%$ la"t

*ermi""i!le attem*t+ a%$ t#at t#e &ate" o( t#e "c#ool

0ol$ !e clo"e$ to #im (oreer+ )aloi" t#re0 t#e

!lack!oar$ era"er i% t#e (ace o( #i" e7ami%er.

)aloi" #a$ to make $o 0it# a "eco%$=!e"t c#oice+

t#e Ecole 1ormale. Bt ee% t#ere #e $i$ %ot (are

0ell. )aloi"' (at#er+ t#e mayor o( Bor&=la=Rei%e+ 0a"t#e tar&et o( clerical i%tri&e" i% t#e to0%. A%

%"cr*lo" *rie"t circlate$ *or%o&ra*#ic er"e"+

"i&%i%& t#em 0it# t#e mayor'" %ame. Mo%t#" o( "c#

*er"ectio% ca"e$ )aloi"' (at#er to lo"e #i"

co%$e%ce a%$ #e !ecame co%i%ce$ t#at t#e 0orl$

0a" ot to &et #im. Slo0ly lo"i%& toc# 0it# reality+

#e 0e%t to Pari". T#ere+ i% a% a*artme%t %ot (ar (rom

0#ere #i" "o% 0a" "t$yi%&+ t#e (at#er kille$ #im"el(.

T#e yo%& )aloi" %eer recoere$ (rom t#i" tra&e$y.

O!"e""e$ 0it# t#e lo"t ca"e o( t#e ,>3/ reoltio%+

a%$ (r"trate$ 0it# t#e "c#ool $irector+ 0#om #eco%"i$ere$ a% a*olo&i"t (or t#e royali"t" a%$ cleric"+

)aloi" 0rote a "cat#i%& letter criticizi%& t#e $irector.

He 0a" i%"*ire$ to $o "o a(ter t#ree $ay" o( rioti%& i%

t#e "treet"+ 0#e% "t$e%t" all oer Pari" 0ere

reolti%& a&ai%"t t#e re&ime. )aloi" a%$ #i"

cla""mate" 0ere ke*t locke$ i% t#eir "c#ool+ %a!le

to "cale t#e tall (e%ce. T#e a%&ry )aloi" "e%t #i"

!li"teri%& letter criticizi%& t#e "c#ool $irector to t#e

.aDette )es +coles* A" a re"lt+ #e 0a" e7*elle$ (rom t#e

"c#ool. Bt )aloi" 0a" %$a%te$#e 0rote a "eco%$

letter to t#e .aDette- a%$ a$$re""e$ t#e "t$e%t" att#e "c#ool to "*eak * (or #o%or a%$ co%"cie%ce. He

&ot %o re"*o%"e.


89/177

Ot o( "c#ool+ )aloi" "tarte$ oNeri%& *riatele""o%" i% mat#ematic". He 0a%te$ to teac# #i" o0%

mat#ematical t#eorie"+ ot"i$e t#e Fre%c# "c#ool"+

0#e% #e 0a" all o( %i%etee% year" ol$. Bt )aloi"

col$ %ot %$ "t$e%t" to teac##i" t#eorie" 0ere

too a$a%ce$ #e 0a" (ar a#ea$ o( #i" time.

Faci%& a% %certai% (tre a%$ $oome$ %ot to !e

a!le to *r"e a $ece%t e$catio%+ i% $e"*eratio%

)aloi" oi%e$ t#e artillery !ra%c# o( t#e Fre%c#

1atio%al )ar$. Amo%& t#e 1atio%al )ar$+ #ea$e$

i% it" *a"t !y La(ayette+ t#ere 0ere ma%y li!eral

eleme%t" clo"e to yo%& )aloi"' *olitical *#ilo"o*#y.9#ile i% t#e )ar$+ )aloi" ma$e o%e la"t attem*t to

*!li"# #i" mat#ematical 0ork. He 0rote a *a*er o%

t#e &e%eral "oltio% o( eatio%"to$ay reco&%ize$

a" t#e !eati(l )aloi" T#eorya%$ "e%t it to Simeo%=

De%i" Poi""o% ,?>,=,>/ at t#e Fre%c# Aca$emy o(

Scie%ce". Poi""o% rea$ t#e *a*er !t $etermi%e$

t#at it 0a" i%com*re#e%"i!le. O%ce a&ai%+ t#e

%i%etee%=year=ol$ 0a" "o (ar a#ea$ o( a%y o( t#e

ol$er Fre%c# mat#ematicia%" o( #i" $ay t#at #i"

ele&a%t %e0 t#eorie" 0e%t 0ay oer t#eir #ea$". At

t#at mome%t+ )aloi" $eci$e$ to a!a%$o%mat#ematic" a%$ to !ecome a (ll=time reoltio%ary.

He "ai$ t#at i( a !o$y 0a" %ee$e$ to &et t#e *eo*le

i%ole$ 0it# t#e reoltio%+ #e 0ol$ $o%ate #i".

O% May 6+ ,>3,+ t0o #%$re$ yo%& re*!lica%"

#el$ a !a%et i% 0#ic# t#ey *rote"te$ a&ai%"t t#e

royal or$er $i"!a%$i%& t#e artillery o( t#e 1atio%al

)ar$. Toa"t" 0ere oNere$ to t#e Fre%c# Reoltio%

a%$ it" #eroe"+ a" 0ell a" to t#e %e0 reoltio% o(

,>3/. )aloi" "too$ * a%$ *ro*o"e$ a toa"t. He "ai$

To Loi" P#ili**e+ t#e Dke o( Orlea%"+ 0#o 0a"

%o0 i%& o( t#e Fre%c#. 9#ile "ayi%& t#i"+ #ol$i%& *#i"


90/177

&la""+ )aloi" 0a" al"o #ol$i%& * a% o*e% *ocket=

k%i(e i% #i" ot#er #a%$. T#i" 0a" i%ter*rete$ a" a

t#reat o% t#e li(e o( t#e i%&+ a%$ ca"e$ a riot. T#e

%e7t $ay )aloi" 0a" arre"te$.

I% #i" trial (or t#reate%i%& t#e li(e o( t#e i%&+

)aloi"' attor%ey claime$ t#at )aloi" #a$ actally "ai$+

0#ile #ol$i%& #i" k%i(e+ To Loi" P#ili**e+ i( #e

!ecome" a traitor. Some o( )aloi"' artillery (rie%$"

0#o 0ere *re"e%t te"tie$ to t#i"+ a%$ t#e ry (o%$

#im %ot &ilty. )aloi" retriee$ #i" k%i(e (rom t#e

ei$e%ce ta!le+ clo"e$ it a%$ *t it i% #i" *ocket+ a%$

le(t t#e cortroom a (ree ma%. Bt #e 0a" %ot (ree

(or ery lo%&. A mo%t# later #e 0a" arre"te$ a" a

$a%&ero" re*!lica%+ a%$ ke*t i% ail 0it#ot a

c#ar&e 0#ile t#e at#oritie" looke$ (or a c#ar&e

a&ai%"t #im t#at 0ol$ "tick. T#ey %ally (o%$ o%e

0eari%& t#e %i(orm o( t#e $i"!a%$e$ artillery. )aloi"

0a" trie$ (or t#i"8#ar&e a%$ "e%te%ce$ to "i7 mo%t#"

i% ail. T#e royali"t" 0erc*lea"e$ to %ally *t a0ay a

t0e%ty=year=ol$ t#ey co%"i$ere$ to !e a $a%&ero"

e%emy o( t#e re&ime. )aloi" 0a" *arole$ a(ter "ome

time a%$ 0a" moe$ to a #al(0ay (acility. 9#at

#a**e%e$ %e7t i" o*e% to e"tio%. 9#ile o% *arole+

)aloi" met a yo%& 0oma% a%$ (ell i% loe. Some

!eliee #e 0a" "et * !y #i" royali"t e%emie" 0#o

0a%te$ to *t a% e%$ to #i" reoltio%ary actiitie"

o%ce a%$ (or all. At a%y rate+ t#e 0oma% 0it# 0#om

#e &ot i%ole$ 0a" o( e"tio%a!le irte E&une

cocjuette )e has e,tage&F* A" "oo% a" t#e t0o !ecame

loer"+ a royali"t came to "ae #er #o%or a%$

c#alle%&e$ )aloi" to a $el. T#e yo%&

mat#ematicia% 0a" le(t %o 0ay ot o( t#e me"". He

trie$ eeryt#i%& #e col$ to talk t#e ma% ot o( t#e

$el+ !t to %o aail.

T#e %it !e(ore t#e $el+ )aloi" 0rote "eeral

letter".


91/177

T#e"e letter" to #i" (rie%$" le%$ "**ort to t#e

t#eory t#at #e 0a" (rame$ !y t#e royali"t". He 0rotet#at #e 0a" c#alle%&e$ !y t0o o( t#e royali"t" a%$

t#at t#ey #a$ *t #im o% #i" #o%or %ot to tell #i"

re*!lica% (rie%$" a!ot t#e $el. I $ie t#e ictim o(

a% i%(amo" coette. It i" i% a mi"era!le !ra0l t#at

my li(e i" e7ti%&i"#e$. O#+ 0#y $ie (or "o triial a

t#i%&+ 0#y $ie (or "omet#i%& "o $e"*ica!le Bt

mo"t o( t#at la"t %it !e(ore t#e $el+ )aloi"

care(lly *t $o0% o% *a*er #i" e%tire mat#ematical

t#eory+ a%$ "e%t it to #i" (rie%$ A&"te C#ealier. At

$a0% o% May 3/+ ,>3@+ )aloi" (ace$ #i" c#alle%&er o%

a $e"erte$ el$. He 0a" "#ot i% t#e "tomac# a%$ le(tlyi%& i% a&o%y alo%e i% t#e el$. 1o o%e !ot#ere$ to

call a $octor. Fi%ally a *ea"a%t (o%$ #im a%$

!rot #im to t#e #o"*ital+ 0#ere #e $ie$ t#e %e7t

mor%i%&. He 0a" t0e%ty year" ol$. I% ,>4+ t#e

mat#ematicia% o"e*# Lioille e$ite$ a%$ *!li"#e$

)aloi"' ele&a%t mat#ematical t#eory i% a or%al.

)aloi"' t#eory 0ol$ "**ly t#e crcial "te* i% t#e

met#o$ "e$ a ce%try a%$ a #al( later i% attacki%&

Fermat'" La"t T#eorem.

nother GictimCac#y'" carele""%e"" a%$ arro&a%ce ri%e$ t#e li(e

o( at lea"t o%e ot#er !rillia%t mat#ematicia%. 1iel"

He%rik A!el ,>/@=,>@6 0a" t#e "o% o( t#e *a"tor o(

t#e illa&e o( Fi%$o i% 1or0ay. 9#e% #e 0a" "i7tee%+

a teac#er e%cora&e$ A!el to rea$ )a""' (amo"

!ook+ t#e 9iscjuisitiones* A!el 0a" ee% "cce""(l i%

lli%& i% "ome &a*" i% t#e *roo(" o( "ome o( t#e

t#eorem". Bt t0o year" later+ #i" (at#er $ie$ a%$

yo%& A!el #a$ to *o"t*o%e #i" "t$y o( mat#ematic"

a%$ co%ce%trate #i" eNort" o% "**orti%& t#e (amily.

I% "*ite o( t#e &reat $i


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(ace$+ A!el ma%a&e$ to co%ti%e "ome "t$y o(

mat#ematic" a%$ ma$e a remarka!le mat#ematical

$i"coery at t#e a&e o( %i%etee%. He *!li"#e$ a

*a*er i% ,>@ i% 0#ic# #e *roe$ t#at %o "oltio%

0a" *o""i!le (or eatio%" o( t#e (t# $e&ree. A!el

#a$ t#" "ole$ o%e o( t#e mo"t cele!rate$ *ro!lem"

o( #i" $ay. 2et t#e &i(te$ yo%& mat#ematicia% #a$

"till %ot !ee% oNere$ a% aca$emic *o"itio%+ 0#ic# #e

!a$ly %ee$e$ to "**ort #i" (amily a%$ "o #e "e%t #i"

0ork to Cac#y (or ealatio% a%$ *o""i!le

*!licatio% a%$ reco&%itio%. T#e .*a*er A!el "e%t

Cac#y 0a" o( e7traor$i%ary *o0er a%$ &e%erality.

Bt Cac#y lo"t it. 9#e% it a**eare$ i% *ri%t+ year"

later+ it 0a" mc# too late to #el* A!el. I% ,>@6 A!el

$ie$ (rom t!erclo"i"+ !rot o% !y *oerty a%$

t#e "trai% o( "**orti%& #i" (amily i% $ire

circm"ta%ce". T0o $ay" a(ter #i" $eat#+ a letter

arrie$ a$$re""e$ to A!el i%(ormi%& #im #e #a$ !ee%

a**oi%te$ *ro(e""or at t#e ;%ier"ity o( Berli%.

T#e co%ce*t o( a% A!elia% )ro* %o0 co%"i$ere$ a

0or$ a%$ 0ritte% 0it# a "mall a+ a!elia% i" ery

im*orta%t i% mo$er% al&e!ra a%$ i" a crcial eleme%t

i% t#e mo$er% treatme%t o( t#e Fermat *ro!lem. A%

a!elia% &ro* i" o%e 0#ere t#e or$er o( mat#ematical

o*eratio%" ca% !e reer"e$ 0it#ot aNecti%& t#e

otcome. A% a!elia% ariety i" a% ee% more a!"tract

al&e!raic e%tity+ a%$ it" "e 0a" al"o im*orta%t i%

mo$er% a**roac#e" to t#e "oltio% o( Fermat'" La"t

T#eorem.

9e)ekin),s ()eals

T#e le&acy o( Carl Frie$ric# )a"" co%ti%e$ t#ro

t#e ce%trie". O%e o( )a""' mo"t %ota!le

mat#ematical "cce""or" 0a" Ric#ar$ De$eki%$

,>3,=,6,4+ !or% i% t#e "ame


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to0% a" t#e &reat ma"ter+ i% Br%"0ick+ )erma%y.

;%like )a""+ #o0eer+ a" a c#il$ De$eki%$ "#o0e$

%o &reat i%tere"t i% or ca*acity (or mat#ematic". He

0a" more i%tere"te$ i% *#y"ic" a%$ c#emi"try+ a%$

"a0 mat#ematic" a" a "era%t to t#e "cie%ce". Bt at

t#e a&e o( "ee%tee%+ De$eki%$ e%tere$ t#e "ame

"c#ool 0#ere t#e &reat )a"" &ot #i" mat#ematical

trai%i%&Caroli%e Colle&ea%$ t#ere #i" (tre

c#a%&e$. De$eki%$ !ecame i%tere"te$ i%

mat#ematic" a%$ #e *r"e$ t#at i%tere"t i%

)otti%&e%+ 0#ere )a"" 0a" teac#i%&. I% ,>-@+ at t#e

a&e o( @,+ Ric#ar$ De$eki%$ receie$ #i" $octorate

(rom )a"". T#e ma"ter (o%$ #i" **il'" $i""ertatio%

o% t#e calcl" com*letely "ati"(yi%&. T#i" 0a" %ot

"c# a &reat com*lime%t+ a%$ i% (act De$eki%$'"

&e%i" #a$ %ot yet !e&% to ma%i(e"t it"el(.

I% ,>-+ De$eki%$ 0a" a**oi%te$ a lectrer i%

)otti%&e%. 9#e% )a"" $ie$ i% ,>-- a%$ Diric#let

moe$ (rom Berli% to take #i" *o"itio%+ De$eki%$

atte%$e$ all o( Diric#let'" lectre" at )otti%&e% a%$

e$ite$ t#e latter'" *io%eeri%& treati"e o% %m!er

t#eory+ a$$i%& a "**leme%t !a"e$ o% #i" o0% 0ork.

T#i" "**leme%t co%tai%e$ a% otli%e o( t#e t#eory

De$eki%$ $eelo*e$ (or al&e!raic %m!er"+ 0#ic#

are $e%e$ a" "oltio%" o( al&e!raic eatio%". T#ey

co%tai% mlti*le" o( "are root" o( %m!er" alo%&

0it# ratio%al %m!er". Al&e!raic %m!er el$" are

ery im*orta%t i% t#e "t$y o( Fermat'" eatio%+ a"

t#ey ari"e (rom t#e "oltio% o( ario" ki%$" o(

eatio%". De$eki%$ t#" $eelo*e$ a "i&%ica%t

area 0it#i% %m!er t#eory.

De$eki%$'" &reate"t co%tri!tio% to t#e mo$er%

a**roac# to Fermat'" la"t T#eorem 0a" #i"

$eelo*me%t o( t#e t#eory o( i$eal"+ a!"tractio%" o(

mmer'" i$eal %m!er". A ce%try


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a(ter t#eir $eelo*me%t !y De$eki%$+ i$eal" 0ol$

i%"*ire Barry Mazr+ a%$ Mazr'" o0% 0ork 0ol$ !ee7*loite$ !y A%$re0 9ile".

I% t#e ,>-?=> aca$emic year+ Ric#ar$ De$eki%$

&ae t#e r"t mat#ematic" cor"e o% )aloi" t#eory.

De$eki%$'" %$er"ta%$i%& o( mat#ematic" 0a" ery

a!"tract+ a%$ #e eleate$ t#e t#eory o( &ro*" to t#e

mo$er% leel at 0#ic# it i" %$er"too$ a%$ tat

to$ay. A!"tractio% ma$e t#e t0e%tiet#=ce%try

a**roac# to Fermat'" *ro!lem *o""i!le. De$eki%$'"

&ro%$!reaki%& cor"e o% t#e t#eorie" $eelo*e$ !y

)aloi" 0a" a &reat "te* i% t#i" $irectio%. T#e cor"e

0a" atte%$e$ !y t0o "t$e%t".T#e% De$eki%$'" career took a "tra%&e tr%. He le(t

)otti%&e% (or a *o"itio% i% Zric#+ a%$ a(ter e

year"+ i% ,>4@+ #e retr%e$ to Br%"0ick+ 0#ere #e

tat at a #i "c#ool (or (ty year". 1o $%e #a"

!ee% a!le to e7*lai% 0#y a !rillia%t mat#ematicia%

0#o !rot al&e!ra to a% i%cre$i!ly #i leel o(

a!"tractio% a%$ &e%erality "$$e%ly le(t o%e o( t#e

mo"t *re"ti&io" *o"itio%" at a%y Ero*ea%

%ier"ity to teac# at a% %k%o0% #i "c#ool.

De$eki%$ %eer marrie$ a%$ lie$ (or ma%y year"

0it# #i" "i"ter. He $ie$ i% ,6,4+ a%$ mai%tai%e$ a"#ar*+ actie mi%$ to #i" la"t $ay.

8in )e Siecle

At t#e tr% o( t#e %i%etee%t# ce%try+ t#ere lie$ i%

Fra%ce a mat#ematicia% o( &reat a!ility i% a

"r*ri"i%&ly 0i$e ariety o( area". T#e !rea$t# o(

k%o0le$&e o( He%ri Poi%care ,>-=,6,@ e7te%$e$

!eyo%$ mat#ematic". I% ,6/@ a%$ later+ 0#e% #e 0a"

alrea$y a re%o0%e$ mat#ematicia%+ Poi%care 0rote

*o*lar


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!ook" o% mat#ematic". T#e"e *a*er!ack"+ rea$ !y

*eo*le o( all a&e"+ 0ere a commo% "it i% t#e ca(e"

a%$ t#e *ark" o( Pari".

Poi%care 0a" !or% to a (amily o( &reat ac#ieer".

Hi" co"i%+ Raymo%$ Poi%care+ ro"e to !e t#e

*re"i$e%t o( Fra%ce $ri%& t#e Fir"t 9orl$ 9ar. Ot#er

(amily mem!er" #el$ &oer%me%t a%$ *!lic "erice

*o"itio%" i% Fra%ce a" 0ell.

From a yo%& a&e+ He%ri $i"*laye$ a *o0er(l

memory. He col$ recite (rom a%y *a&e o( a !ook #e

rea$. Hi" a!"e%tmi%$=e$%e""+ #o0eer+ 0a"

le&e%$ary. A Fi%%i"# mat#ematicia% o%ce came all

t#e 0ay to Pari" to meet 0it# Poi%care a%$ $i"c""

"ome mat#ematical *ro!lem". T#e i"itor 0a" ke*t

0aiti%& (or t#ree #or" ot"i$e Poi%care'" "t$y

0#ile t#e a!"e%tmi%$e$ mat#ematicia% *ace$ !ack

a%$ (ort#a" 0a" #i" #a!it t#root #i" 0orki%&

li(e. Fi%ally+ Poi%care *o**e$ #i" #ea$ i%to t#e

0aiti%& room a%$ e7claime$8 Sir+ yo are $i"tr!i%&

me *o% 0#ic# t#e i"itor "mmarily le(t+ %eer to

!e "ee% i% Pari" a&ai%.

Poi%care'" !rillia%ce 0a" reco&%ize$ 0#e% #e 0a"

i% eleme%tary "c#ool. Bt "i%ce #e 0a" "c# a

%ier"ali"ta re%ai""a%ce ma% i% t#e maki%i""*ecial a*tit$e (or mat#ematic" $i$ %ot yet

ma%i(e"t it"el(. He $i"ti%&i"#e$ #im"el( 0it# #i"

e7celle%ce i% 0riti%& at a% early a&e. A teac#er 0#o

reco&%ize$ a%$ e%cora&e$ #i" a!ility trea"re$ #i"

"c#ool *a*er". At "ome *oi%t+ #o0eer+ t#e

co%cer%e$ teac#er #a$ to catio% t#e yo%& &e%i"8

Do%'t $o "o 0ell+ *lea"e...try to !e more or$i%ary.

T#e teac#er #a$ &oo$ rea"o% to make t#i" "&&e"=

tio%. A**are%tly+ Fre%c# e$cator" #a$ lear%e$

"omet#i%& (rom t#e mi"(ort%e" o( )aloi" #al( a

ce%try earlierteac#er" (o%$ t#at &i(te$ "t$e%t"o(te% (aile$ at t#e #a%$" o( %i%"*ire$ e7ami%er".

Hi" teac#er 0a" &e%i%ely 0orrie$ t#at


96/177


97/177

FERMAT'S LAST THEOREM

!o)ular 8orms

Poi%care "t$ie$ *erio$ic (%ctio%"+ "c# a" t#e "i%e"

a%$ co"i%e" o( Forier%ot o% t#e %m!er li%e a"

Forier #a$ $o%e+ !t i% t#e com*le7 *la%e. T#e "i%e

(%ctio%+ sin #- i" t#e ertical #eit o% a circle 0it#

ra$i" , 0#e% t#e a%&le i" 7. T#i" (%ctio% i" *erio$ic8

it re*eat" it"el( oer a%$ oer eery time t#e a%&lecom*lete" a mlti*le o( it" *erio$+ 34/ $e&ree". T#i"

*erio$icity i" a "ymmetry. Poi%care e7ami%e$ t#e

com*le7 *la%e+ 0#ic# co%tai%" real %m!er" o% t#e

#orizo%tal a7i" a%$ ima&i%ary o%e" o% t#e ertical o%e

a" "#o0% !elo0.

5ere, a periodic unction could be conceived as having a periodicity

both along the real a&is an) along the imaginary a&is. +oincare went

even urther and posited the e&istence o unctions with a wider

array o symmetries. These were unctions that remained

unchanged when the comple& variable z was changedaccording toOz20000000000000HjEaD$bIcD$)F* 5ere the elements a- b- c,

)- arranged as a matri&, ormed an algebraic grou'*This means


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t#at t#ere are i%%itely ma%y *o""i!leariatio%". T#ey all commte 0it# eac#

ot#er a%$ t#e (%ctio% i" invariant %$ert#i" &ro* o( tra%"(ormatio%". Poi%carecalle$ "c# 0eir$ (%ctio%" atomor*#ic

(orm".T#e atomor*#ic (orm" 0ere ery+ ery

"tra%&e creatre" "i%ce t#ey "ati"e$ma%y i%ter%al "ymmetrie". Poi%care

0a"%'t ite "re t#ey e7i"te$. Actally+

Poi%care $e"cri!e$ #i" re"earc# a"(ollo0". He "ai$ t#at (or (tee% $ay" #e

0ol$ 0ake * i% t#e mor%i%& a%$ "itat #i" $e"k (or a co*le o( #or" tryi%&

to co%i%ce #im"el( t#at t#e atomor*#ic(orm" #e i%e%te$ col$%'t *o""i!ly e7i"t.

O% t#e (tee%t# $ay+ #e realize$ t#at #e0a" 0ro%&. T#e"e "tra%&e (%ctio%"+

#ar$ to ima&i%e i"ally+ $i$ e7i"t.Poi%care e7te%$e$ t#em to ee% morecom*licate$ (%ctio%"+ calle$ mo$lar

(orm". T#e mo$lar (orm" lie o% t#e**er #al( o( t#e com*le7 *la%e+ a%$

t#ey #ae a #y*er!olic &eometry. T#ati"+ t#ey lie i% a "tra%&e "*ace 0#ere

t#e %o%=Ecli$ea% &eometry o( Bolyai a%$Lo!ac#e"ky rle". T#ro a%y *oi%t i%t#i" #al(=*la%e+ma%y li%e" *arallel to

a%y &ie% li%e e7i"t.


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Poi%care le(t !e#i%$ t#e "ymmetric

atomor*#ic (orm" a%$ t#eir ee% morei%tricate mo$lar (orm" a%$ 0e%t o% to

$o ot#er mat#ematic". He 0a" !"y i%"o ma%y el$"+ o(te% i% a (e0 o( t#emat o%ce+ t#at #e #a$ %o time to "it a%$

*o%$er t#e !eaty o( #ar$ly ima&i%a!le+i%%itely "ymmetric e%titie". Bt

%!ek%o0%"t to #im+ #e #a$ "t "o0e$a%ot#er "ee$ i% t#e &ar$e% t#at 0ol$

ee%tally !ri%& a!ot t#e Fermat"oltio%.

n Une#'ecte) 2onnection with a 9oughnut

I% ,6@@+ t#e America% mat#ematicia%Loi" . Mor$ell $i"coere$ 0#at #e

t#ot 0a" a ery "tra%&e co%%ectio%!et0ee%

The very strane !o"#$ar %or!s are sy!!etr&c &n !any 'ays '&th&n th&s s(ace. The sy!!etr&es are o)ta&ne" )y a""&n a n#!)


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t#e "oltio%" o( al&e!raic eatio%" a%$to*olo&y. T#e eleme%t" o( to*olo&y are"r(ace" a%$ "*ace". T#e"e "r(ace"

col$ !e i% a%y $ime%"io%8 t0o$ime%"io%"+ "c# a" t#e &re" i%

a%cie%t )reek &eometry+ or t#ey col$!e i% t#ree=$ime%"io%al "*ace+ or more.

To*olo&y i" a "t$y o( co%ti%o" (%c=tio%" t#at act o% t#e"e "*ace"+ a%$ t#e

*ro*ertie" o( t#e "*ace" t#em"ele". T#e*art o( to*olo&y 0#ic# co%cer%e$

Mor$ell 0a" t#e o%e o( "r(ace" i%t#ree=$ime%"io%al "*ace. A "im*lee7am*le o( "c# a "r(ace i" a "*#ere8

it i" t#e "r(ace o( a !all+ "c# a" a!a"ket!all. T#e !all i" t#ree=$ime%"io%al+

!t it" "r(ace a""mi%& %o $e*t# i" at0o=$ime%"io%al o!ect. T#e "r(ace o(t#e eart# i" a%ot#er e7am*le. T#e eart#

it"el( i" t#ree $ime%"io%al8 a%y *lace o%t#e eart# or i%"i$e it ca% !e &ie% !y

it" lo%&it$e o%e $ime%"io%+ it"

latit$e a "eco%$ $ime%"io%+ a%$ it"$e*t# t#e t#ir$ $ime%"io%. Bt t#e"r(ace o( t#e eart# %o $e*t# i" t0o=$ime%"io%al+ "i%ce a%y *oi%t o% t#e

"r(ace o( t#e eart# ca% !e "*ecie$ !yt0o %m!er"8 it" lo%&it$e a%$ it"

latit$e.T0o=$ime%"io%al "r(ace" i% t#ree=

$ime%"io%al "*ace ca% !e cla""ie$accor$i%& to t#eir &e%". T#e &e%" i"t#e %m!er o( #ole" i% t#e "r(ace. T#e

&e%" o( a "*#ere i" zero "i%ce t#ereare %o #ole" i% it. A $o%t #a" o%e

#ole i% it. T#ere(ore t#e &e%" o( a$o%t mat#ematically calle$ a tor"

i" o%e. A #ole mea%" a #ole t#at r%"com*letely t#ro t#e "r(ace. A c*0it# t0o #a%$le" #a" t0o #ole" t#ro

it. T#ere(ore it i" a "r(ace o( &e%"t0o.

A "r(ace o( o%e &e%" ca% !etra%"(orme$ !y a co%ti%o" (%ctio%

i%to a%ot#er "r(ace o( t#e "ame &e%".T#e o%ly 0ay to tra%"(orm a "r(ace o(


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aczel, amir d - fermat's last theorem_uc

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