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w!r was n!t as highly regarded as it deser ed t! e ecause he ga e n! e p!siti!ns !$ his meth!ds,

and he used the aw ward n!tati!n !$ Fran?!is @i te, which had een made ! s!lete y escartes;

n!tati!ns used in his Gomtrie %

Fermat;s ma !r mathematical interest was in num er the!ry% B!me !$ the m!st alua le !$ his results

were disc! ered a$ter his death !n l!!se sheets !$ paper !r in the margins !$ !! s he had read and

ann!tated% 'is $riends $eared that since he had n!t pu lished anything his w!r w!uld e $!rg!tten

a$ter his death% 'is s!n 8lement Bamuel c!llected his $ather;s letters, mathematical papers, and

c!mments written in !! s, a t!tal !$ s!me :000 mathematical items% +he Opera mathematica !$

Fermat were pu lished in tw! !lumes in 1670 and 167C% t was in the margins !$ his c!py !$ the n!w

l!st 8laude 4achet;s translati!n !$ i!phantus; Arithmetica that his $am!us "*ast +he!rem& appears%

'e claimed that there are n! p!siti e integers x, y and z such that, xn D yn E z n, $!r integers n 2%

Fermat went !n t! sayG " ; e $!und a remar a le pr!!$ !$ this $act, ut there is n!t en!ugh space in the

margin t! write it%&

n 1CC> Andrew Hiles success$ully pr! ed the the!rem that had $ascinated many pr!$essi!nal and

amateur mathematicians in the inter ening years% As t! Fermat;s claim !$ ha ing $!und "a remar a le

pr!!$,& it is unli ely% 'e may ha e disc! ered a meth!d $!r pr! ing the cases when nE: and nE> and

ecame c!n inced that similar pr!!$s w!uld w!r in the general case% An!ther !$ Fermat;s pu liciIed

c!n ectures turned !ut t! e $alse% 'e asserted his elie$ that num ers !$ the $!rm 1 m!re than 2 raised

t! the p!wer 2 n were prime num ers% 'e seems t! ha e een c!n inced when this was sh!wn t! e the

case $!r n 5% A century later Kuler dem!nstrated the s! called Fermat number F(n) was n!t prime

when n E 5, in $act 2 :2 D 1 E >,2C>,C67,2C7 E 6>1 6,700,>17%

*ater 8arl Friedrich

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p!lyg!n with n sides is c!nstructi le with straightedge and c!mpass i$ n is a prime Fermat num er !r a

pr!duct !$ di$$erent Fermat primes% Fermat;s claim !$ ha ing $!und a pr!!$ !$ his "*ast +he!rem& was

!nly a pri ate c!mment, which wasn;t c!rresp!nded t! !ther mathematicians n!r did he !$$er it as a

challenge t! !thers as he !$ten did with results he had already pr! ed% #erhaps Fermat disc! ered a$law in the pr!!$ he elie ed he had, and ne er cr!ssed !ut his c!mment ecause he ne er e pected

any!ne t! see it%

Fermat was the $irst t! use a meth!d !$ pr!!$ called the "in$inite descent%& t is a particular type !$

pr!!$ y mathematical inducti!n% A typical applicati!n is sh!w that n! p!siti e integer e ists with a

certain pr!perty% Assume the c!ntrary, that s!me p!siti e integer x has the pr!perty% 3e t, deduce that

there is s!me p!siti e integer y x that als! has the pr!perty% .epeat this argument inde$initely thus

in$initely descending thr!ugh all p!siti e integers% +hen !ne must sh!w that this in$inite descent

implied y ha ing a wh!le se uence !$ s!luti!ns that are e en smaller, y !ur ch!sen measure, is

imp!ssi le% +his is a c!ntradicti!n s! n! p!siti e integer e ists with the gi en pr!perty% n a letter

u!ted in Heil;s Number Theory , Fermat descri ed his use meth!dsG

"As !rdinary meth!ds, such as are $!und in !! s, are inade uate t! pr! ing such di$$icult

pr!p!siti!ns, disc! ered at last a m!st singular meth!d L which called the in$inite descent%

At $irst used it !nly t! pr! e negati e asserti!nsL +! apply it t! a$$irmati e uesti!ns is

much harder, s! when had t! pr! e MK ery prime !$ the $!rm > n D 1 is the sum !$ tw!

s uares,& $!und mysel$ in a s!rry plight (en elle peine)% 4ut at last such uesti!ns pr! ed

amena le t! my meth!ds%&

uring the peri!d $r!m 16>: t! 165>, Fermat;s c!rresp!ndence with !ther mathematicians ceased in

part ecause !$ a ci il war in the c!untry greatly a$$ecting +!ul!use% Further the plague struc the

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