Scientific American Volume 239 issue 4 1978 [doi 10.1038%2Fscientificamerican1078-104] Edwards,...

8/10/2019 Scientific American Volume 239 issue 4 1978 [doi 10.1038%2Fscientificamerican1078-104] Edwards, Harold M. -- F

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For 300 years mathematicians have ied withou success to proa theorem Pierre de Fermat asserted he could prove: o nth powecan be the sum o two other nth powers, where n is greater tha 2

he 7thcetury rech atheatca Perre de erat was thefather of oder uber theory

the brach of atheatcs that dealswth the proertes of whole ubers

Le ay other scholars of hs te hestude the classcs of atquty u-ber theory he drew sprato froDophatus the Gree atheatcawhose Aithmetica had bee redscov-ered by Europeas the d16th ce-tury erat ade extesve argalnotes hs copy of the boo ad af-ter hs death 1665 hs so publsheda new edto of Aithmetica cludgerats otes Oe of the otes hasbecoe oe of the ost faous stateets the hstory of atheatcs

Next to a proble o dg squaresthat are sus of two other squares (forexaple 25 equals 9 pus 6 eratwrote (to traslate fro the Lat Othe other had t s possble for acube to be the su of two cubes afourth power to be the su of twofourth powers or geeral for ayuber that s a power greater tha thesecod to be the su of two le powersI have dscovered a truly arvelousdeostrato of ths proposto thatths arg s too arrow to cota"The proposto has coe to be calederats last theore ad although theeeest atheatcas of the past

three cetures ave tred to prove t osuch deostrato has bee fouderats last theore reas oe ofthe great usolved probles of oderatheatcs

Dd erat eally have a arvelousproof" of the theore He was certalya prodgous atheatca helpg toestabsh the dscples of aalytc geoetry (wth Descartes tesalcalcuus (wth Lez ad Newto adprobabty theory (wth Pascal He wasot a atheatca by professo buta urst at Tououse the south of

race Hs extesve partcpato the telectual lfe of hs te was coducted etrely through hs prvate cor-respodece wth other scholars theaor Europea ceters of learg

t is charg to th of a provcal

d dd

17thcetury rst outsartg the bestatheatcal ds of three ceureswth hs theore but the facts suggestthat he dd ot have a proof Except forthe faous ote the arg of Do

phantus boo there s o eto of aproof of the theore ay of eratswrtgs that have survved He dd stateelsewhere that he could show t was possble to nd solutos to the equa-tos x3 + y3 z3 ad X4 +y4 Z4. fhe dscovered a vald proof of the fultheore (that xn + yn zn s possble for postve whole ubers x y zad where s greater tha 2 t ssurprsg that he dd ot eto t toot sees ost lely that at the te hewrote the ote erat had a dea for aproof he later realzed was faulty Hsotes were alost certaly ot ted-ed for publcato ad there ay havebeen no occaso for h to go bac addelete or aed the ote

t s of course ore exctg to beleveerat dd have a rgorous proof of thetheore ad t s ust possble that hedd ether case the theore hasplayed a portat role the developet of uber theory Soe ofthe greatest creatos of atheacathought have bee propted by thestudy of the theore ad he techquesthat have bee developed the eort toprove t have cotrbuted to the soluto

of ay other problesally the hstory of erats las

theore provdes a excee ustrato of the real ature of atheatcal qury atheatcas are ofte

ase Ho s t possb to do research atacs" dou!t that physcstsor astrooers or bgss are everased such a questo bt to ay peo-ple atheatcs sees o be sch a cut-

addred subect that wor t shouldaout to othg ore tha a orderlyrecordg of the facts Of course othg could be furher fro the truth natheatcs as ay other eld uaswered questos are everywhere adthe dculty atheatcas face dg questos they ca aswer ooes they caot t s dcult howeverto llustrate the pot for a omatheatca because the forulaton terestg atheatcal questos ualy calls for specalzed termolog bacgroud erat's last theorem i rare excepto to ths rue

nne s

Part of the fascato of erat'slast theore s due to the fact that thetheore s uusually sple to state aduderstad t s possble to d pos-tve whole ubers x: y z ad forwhch s greater tha 2 a the equato xn + yn zn hods Noatheatcas usually approach the theore what sees to be a very reasoabe wayby tryg t ou" Cosder the casewhere s equal to that s the case

that cossts provg tha x3 + y3 z3has o solto The cubes of the rst 0postve tegers are 1 8 27 64 252 6 4 52 729 ad 000 t s otct o see that oe of these u

PIERRE DE FERMAT is caled the father of number theory, the branch of mathematics thatdeals with the properties of the whole numbers. Born n near Toulouse, Fermat spent allhs life n the south of rance, far from the great European centers of learning. He worked as ajurist, not as a professional mathematician, and none of his mathematical works was publsheduntl after his death. Hs extensive participation n the mathematics of his time wa conductedentirely through his private correspondence with other scholars. Fermat formulated many

challenging and perceptve theorems that were not proved until long after his deah. By only one of these theorems remained to be proved. It has come to be called Fermat's last theo-rem: There are no solutions in whole numbers to the equation Z where s great-er than Fermat's last theorem is oe of the most famous unsolved problems of modern mathematics Ths painting of Fermat is in the collection of the Acadmie des Sciences, Inscriptionset Belles Lettres de Toulouse; it is reproduced here with the kind permisson of Robert Gillis.

1978 SCIENTIFIC AMERICAN, INC


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A BABLONIAN CLAY CUNEIFORM TALET dating fomabout 1500 oe of the oldt-known documen dealing withnumbe theoy. The tablet lis (in slightly disguised fom) seveal setsof Pythagoean tipl o distinct positive whole numbe x, y andz, fo whic x2 + y = Z fo example 4961 6480 and 8161 Femat fomulated his last theoem while he was considering a poblem concerning Pythagorean tipl (see illustration on next pge).Any tiple povid an immediate poof that the theoem is not tuewhen the exponent n is equal to Pyhagorean tiples are named

fo thei elation to the Pythagorean therem which states hat in aight tiangle the squae of the hyptenuse is equal to he sum f thesqua of the othe two sides. It is not easy to nd tries by iaan eo paticulay those that involve age numbers and so eBabylonians pobably had a method f nding hem. It seems mslikely that thei inteest stemmed frm the gemeic aicais the sets of numbers and that therefre they knew f the yhagrean theoem 1000 yea befoe Pyhagras. The ablet is a f thePlimpton ollection in the utler ibrary a umbia iey

ers can e expressed as the sum of twoother cues For example. f 512 wereequa to the sum of two other cues. the

cues would e smaler than 512 andwould precede t on the lst of cuesThe cues 216 and 343 are good cand-dates. ut ther sum s 559 whch s oolarge The next smaler sum. 216 plus216 equals 43 and not 512 Hence 512s not equal to the sum of any two cues

t s easy to verfy y ths procedurethat any partcular cue s not the sumof two other cues The requred cal-culatons could e carred out at hghspeed on a computer. and t could eshown very qucky that no cue wthfewer than. say. 10 dgts s the sum oftwo other cues There are. however. annnte numer of cues to e tested. sothat wth ths procedure even the largestcomputer coud never resolve the queston of whether any cue can e the sumof two other cues

Once nonmathematcans have seenthat the mposlty of x3 y3 z3cannot e proved smply y tryng t."

they may swng to the opposte extremeand ask how such a statement can evere proved The answer s that t can eproved y the method of reasonng to acontradcton One assumes that there sa soluton to the equaton. and one de-duces from ths assumpton a statementknown to e false Arrvng at a fasestatement. or contradcton. shows thatthe orgnal assumpton was false andtherefore that there can e no soluton

More speccally. n the case of state-ments. such as Fermats last theorem.that are concerned wth postve wholenumers a proof y contradcton oftentakes the form of a proof y nnte de-scent Fermat mantaned he had nvent-ed ths method. whch he sad was theass of all hs proofs n the eld of num-er theory n a proof y nnte descent

one shows that f a soluton n postvewhole umers to the equaton n queston s gven. t can e used to produce a

smaller soluton. also n postve wholenumers The same argument thenshows that the smaller soluton can eused to produce a stll smaller one. andthe process can e repeated ndentelySnce the solutons are al postve wholenumbes. however. t s ovously mpossle to nd ncreasngly small solutons ad nntum Hence there ca no soluton at all

a

n all ermats wrtgs on umertheory that have een preserved there sonly one proof. found n another of themargnal notes n Dophantus Aithmetica t concerns Pythagorean tragles.or rght trangles wth sdes of tegerlength (They derve ther ame fro



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their relation to the Pythagorean heorem, whch states tha the sides x and and the hypotenuse z of right rianglere related by the equation x + =z Fermat proved that the area ofsuch a triangle cannt be a square: I x

and z are ositive integers and x + = z then (!x is not the square

of an integer. It s easy to show that ether x or mus be even and thereorethat (!x must be whole nuer

Ferat used the method o nnie descent in the proof Speciclly, he gavean exli method by which, given ositive whole numers x z and u thatsaisy the equations + y = z and

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It is interesing an raly nt cincidental that h at hat the ara o aPythagrean rianl annt e a squareimeiately ilis at X4 + 4 = Z4has no slutin, r hat ra's lasttheorem is true r h ase whre neqals A sile ut ingenious tricksues to connect he two rositions:assume that X4 + 4 = Z4 or sme ositive integers x and z and set a equal o

4 equal to equal to Z4 + X4and equal to Then the silealgeraic dentity (r + S= r + rs + Simplies that a + equals Z4 4 +X4Z4, or Z8 X4Z4 + x8 + X4Z4, orZ4 + X4)2, which is equa to Furhrmore, (!a is equal to (1!y4xzor xz whch is equal to Thusa + = and (1!a = whihFermat roved was iossile Henethe original assumtin, that X4 +4=Z4 has a solution, must be incorrect,and Ferat's last theorem is estalishedor the case n equals In essene then,Fermat hisel roved his last theorefor the case o ourth powers

This prof also esalshes that Fermat's last therm is true whenver n sa multile o , eause i n is equal to

or soe osiive ineger thenx + y = z ilies hat (k4 + (yk4=(zk4 which is iossile cause aourth oer cannot be the s o twoother fourth owrs In exactly the saeay, i the theorem can e rove rany given exnen m then t is trueor all mulils o m Thrre sineevery whole nur n gratr than is ivisile either y an od rime (aprime nuer ohr than or y itsues to rve Ferat's last thrm in all cases where the exonnt isan odd rie nuer

Ferat mainain he could rvethe therem in the case n equals 3 butno ulished roo o the issiiliyo ning a soluion or x3 + 3 = z3aeared until aout 100 years larThe r hih was the work o the



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Si aaiian Lnad Eudid v av a iu a

lr r

Eu vd a qua 3 inni dn ing a d i i i i div an givn uin and z

quain + z a n uin Yand Z u ta Z i an z Td i t ng xain in dtai u in uin aking auain nning vaiu caaiti and z Eul dd t diving a al utin t n ving t ling iin I pand q a raiv i intg (ity av n cn a diviga tan 1) and p2 + 3q2 i a utn t mut b ing a and b uthat p qual a3- 9ab2 and q qual3a2b 3b2. T itin i qui

t and an vd aing i vaiain d und in Eu' uid k Ini inan wv Eu a nvl t agun tainvvd induing nu a + b v, a and b a ing

T undtand Eu und

nu a + b v uul xand txpin (a + b )3 It i qualt a3 + 3a2b- 9ab2- 3 (a- 9ab2) + (3a2b- 3b) , op + q p and q a dnda ty a in t nuin o tpiin In t d t cnluin t piin at tat(a + b) p + q v N tppitin aum tat p2 + q2 ia cu I p2 + q2 i rrittn as + qY)(P- q), thn t piin an b ratd I (P + q)(P- q) i a cu tn p + q

LEONHARD EULER proved Fermat's ast theorem for the case of the rst odd prime exponen equals This painting, made in about 174, shows the Swiss mathematician during his

lifetime. Bfore Euer's proof Fermat's ast theorem had been estabished (by a direct appication of Fermat's own work) in ony one case: equals 4. It is easy to show that if the theorem istrue when equals some integer then it is ue when equals any multipe of Since everyinteger is divisible either by 4 or by an odd prime, it remained only to prove the theorem in thecas where is an odd prime. The rst successfu eort in this directin was Euer's, but athough his proof was ingenious, it had a serious aw The aw was easiy corrected i the proofof the case equals 3, but it tured out to be a cetra issue in many ater attac theorem

2

u a u a i u av (a + b Y3 iga and b Tu inuin nu a + b ak i i t i iin u i and naua

T nu a + anu u ik a ing In i a ading ua

ing o uiing an nu ith ym giv an nu in t a dividing nuuually d n xa 4 nt divid 5 in ig(tat i n ing an uii 4t giv 5 and givn t nu m a + b v i i n uua ibl t nd a tid nu a tat i ti quin T iiaii t to m ld Eu a an inovaiv and in in i H appid an aid t intg t nu a +

T ty t ing a Eu ndd r i p i div t niqu atiain ing inprim A piiv ing an itn a a pdut pi a in juon ay Fr xa 1 i qua 2 2 1 and n pi nu than o 1 divid 1 vn Uniquactoiati imi t ig ty o t intg A du ativly pim iiv ing an a cb nly i ac t ing i acub xal au a and a rlativy i and a i quato 1000 103 Exd a a u

o pim a 1000 qua 3 X 53T atiain 1000 a aindby pitting h a in u xa ( X 5 5 X 5 0 50 I iing u dnin u a a ta u a aivl i n a ' u g inn u and al 5 u g i Hn n i vau t a and a 3 X 3and 1 al i a u iiat rduc w aiv i iiv ing an an i a ing i a w

Eu aud a i ing a a a num a + b an iin wi i agunt T iin a ai p and q a aiv i a(P + q Y)( - q) i a utn p + q 3 i a u Eu d a i p and q a aipi n p + q Y a qa a aiv i B ni t ing u aiv i nu a + b an a u i nu a un auin a ( + qv)(P Y i a u ii a q i a u and iii vd

T agu ai g Eu i di nui i



10/20

at raoig y aalogy to t aritmti o t itgr do ot otituta valid roo Argumt y aalogy arextrmly uggtiv ad as t itoryo matmati dmotrat a yildul ida ut ty aot tak aroo o aytig It is artilarly urriig tat Eulr wa ot mor irum i i o aaogy a

altoug t umr + b doav may rorti i ommo witt itgr tr ar ao may rorti i wi t two umr ytmdir For xaml t itgr ava atural ordrig 1 ut t umr + b doot Tr i oly o way to ur teumr + b ar wit te itgr t rorty tat t rodut otwo rlativly im umrs i a cueoly if a o t umr i a ad tat i to rov ty do

e a er Ev t t matmatiia ow

evr um to t tmtatio oroo y aalogy rom tim to timot failig to uitly ritialo a argumt au ty kow teoluio it ra i corrt Ttmtatio i artiularly trog wt argumet a t aalig imliity of Eulr It i quit poila Eulr gltd to more rigorou a kw rom or cosidratio that t al coluio + q

( + b

3 wa or

rt Log or uied i rooo te a n quals 3 o Frma' laorm Eulr workd o otr urovd artio o Frmat' orig t rrtatio o umr i tform X2 + 3y2. I artiular rovdFrma's artio tat vry rimumr tat i o mor ta a multil o 3 qual 3n + 1) a a uiqurrtatio a a quar l trtim a quar ( qal x2 + 3y2); orxaml qal X 3 + 1 ad 2 +3 X 12 T tiqu Eulr dvod i t roo o ti torm ar

eaily alid to rov t rooitioi te a n qual 3 Frmat' attorm It i oi Eulr ralid ould rov t rooitio wit t talid tiqu ad tror didot ujct i mor uuual roo tosuitly arul rutiy

I te arlir roo Eulr wa xrmly autiou aout mloyg quioal argumt For xam a irmdiat t i t roo rquir aproof tat i ad b ar rlativly rimitgr t evry odd rim ator oa2 + 3b2 ca writte i t ormc2 + 3d2 I ti ita ulr did othave t dagrou aurae tat tstatm to rovd wa orrt adhis dmotratio i a modl of larityd rigor It i urio owvr tat hxrie wit tis proof did ot alrtim o he urliaiity of his lar arg-

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ment The argument by aalogy can beused to prove that every od pre factor of a2 + 3b2 s of the form c2 3d2ut it an aso be used to prove tatever odd prime factor of a2 + 5b2 s ofthe form c2 + d2, which s false (Notethat 42 + X 12 equas 21 an that ether f he factors of 21 3 and 7 can ewritten as the sum o a square and ve

mes a square) Such onsderatonsma have prompted the rigor that caracterizes Euer's earier proof He hadevidenty forgoten them when he rerned to he sujet many years ater

er' apse in rigor is ose related i grea trengts. name his exrarinar imaginatin and inventiveness is iit ee new nnectionsan ormate prems in rigia anderetive was made is work a greatsoure inspirtin or generation fmatemaicians is aorng f enumer a + b nt primes yanaog he atrizatin f te inte

ger erainy shed great ngenuit Ia ug is ppliationof te ideain is pr te ase euals 3 fFermat' as erem was prematureater nts wd rve he idea t en inspired ne Ideed he aw in ur rgumenthe at hat proertderived rm th nique acorizaton e ineger de nt necessari hor numer ssems resembing the inteer in ter wawoud urn ou t e entra issue in mre sophisticated investigaions he heorem

r ss

Euer's proof needed some patchingbut it esseniay estalished Fermat'sast teorem for te ase where equasthe rst dd prime 3 In the 1820's Gustav Lejeune Dirichet and AdreMarieLegendre proved the teorem for thecase where equas the next pre .Ther metod of prof was bascall anextension of the one Euler had used nhe proo he ase equals 3 andthe anaoge of the ruca equatonp + (a + b)3 was the

eqation p +

(a + b ) As increases he equatons ecounteredin this tpe of proof ecome much moreompex and the metod ails to work)For he ase equas 5 n order o provethat p + is a fth power it must eassmed nt ony that p2 - 52 s a fthpower and p and are relatvely primeas n the case equals 3 but also thatthey have opposte parity (tat one isodd and the other is even) and that isdivsble b 5 Drichets proof of thisfact was based on a completel rigorous study of the numbers of the formx2 5 The proof odeled on oherworks of Eulers including hs rgorousstudy of numbers of the form x2 + 32 .and on works by JosephLous Lagrangeand Car Fredrch Gauss, did not rey ian way on analogy to the factorizatonof the integers

Soe 1 years passed before GbrielLa proved Fermats ast teorem forthe case of he next prime: The proofwas a great acomplshment but it didnot boe well for future proofs ecauset was ong dcut and ose oundto the number There was tte hopethat the proof coud be applied t thenext ase equas 11 or to an ther

ases of the teorem I seemed henthat substanta progess woud nt emade i the study of Ferat' as herem unt a new approac o t prem was disovered

Lam himsef proposed suc an approa n 18 He tred to prve hegenera theore. introducing ompexth root of unty: a compex numer for whc equals 1 but k does notequal 1 for any positve integer esshan The idea was no ne In theprecedng century Lagrange had pointed out that introducng ino te studyo Fermat's ast teorem makes it possi

ble t split + nto facorseac ne onaining and to he rstpower (It is usuay easier to deal withlower powers of varabes)

T obtain the factorizatio note that2 are the roots or sotions. f te equation X 1 and sy he fudamenta theorem f aera X 1 (X 1 )(X )(X 2) X No set X equa xand mutip both sides of the equation Snce onl the cases where iodd are under onsideraton the resutngequationis + ( + )( + y)

( + 2y) . ( + )

m grs

Each of the fators of + n is aumber of the form ao + al + a22 n_n

were ao are ntegers Toda the numbers of thistypethe nubers in the sstem madeup of the ntegers and powers of are called the cclotoc integers. beause the th root of unity s cosely related to the problem of divdnga crcle io equal pars (The om

plex number can be nterpreted as apoint on a crle of uit radius wosecenter is at the orign of the ompexplae; he arc of the ircle tha ies etween 1 and s th of the entirecircle) Like the numbers a + b the cycotomic integers form a number sstem resemblng the integers inthat addng subtracting or mutplngwo cclotoc integers resuts in a ycotomic integer ut dvdng usuaydoes not

ams reatment of the arthmeticof te cyclotoic integers resemesEulers treatment of he numbers

a b although it ma have beena nependent inventon Given thefacorzaion of + n nto ccotomicegers a proposed appling thefac" tat the product of relatvelpre ubers (here by numbers he



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H w alther

p

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T athr of he new heor wasrnt dar Kmmer Several earsarir Kmmr had realized tat orprm in nmr tr c as Ferma a thrm t imprtant proprt rdinar intgr i heir niqeactrizatin int prime and so he haaemped o prove ha he proper appi t th ccloomic inegers as welWhat proved however was haniqe facorizaion sal does nohold r hose nmbers (He had pb-lished he nding in 1844 b in a rahr scur pac ) As Kmmer conined to or on he ccloomic inegershowever it became clear o him hhe fll power of niqe facorizaion,which cold no be obained in he c-cloomic inegers, was no rea nec-essar His heor of 1847 showed hahere ws a wa o modi he concepof niqe facorizaion so ha he moied version cod be appied o proesome sbe and sefl properies o he

ccloomic inegershe basis of Kmmers heor was

he inrodcion of wha he cae ieprime facors ino he arihmeic o c-cloomic inegers, in a manner some

a nau t ntrductin aar mr r nt rmi rinar ntr a i aratr Kummr a mr xpt t a trtr m t m imprtantprprti riv rm niq actrizatin tmi intr and ttr nmr tm c a t nm

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+ a ari in prvint ar as rmat' ast thrm or da actrizatina nqesionabl n th majracimnt 19tcntr matmatic Trug a ratr bizarr utin trming Kmmr' idaprim numr and rtain lasses numr hat ar ratd t tm arn ad ida Tda ida t-r i a parat ranc mahema-ic ic i rtr tstimn t tarracing imprtanc Kmmr'ida i r istrat a srange facat mahemaical research namel

tat it i impossible o predic in adanc ic in invesigaion wilea t sefl discoveries His sd oxtrmel pure r heoreical qes-tin in t d nmer theor edim t rmuat cncpt unforsa vale and versaili for mahema-ic i general

Kmmer' heor led in pariclar tot ratst advance tat a ever beenmad n t tud rmat' ast trm On a ears earlier th proofsfor t cas qa and n eqals 7ad n considered major accomplish

mnt b n 1847 Kmmr a able toprv t heorem re for all prime eponens smaller han 3, hereb ofcorse esablishing he heorem for aeponens smaller han 3 Moreover,he came close o proving he heoremfor all prime eponen smaller han100 Onl he eponens 37 5 and 67eded his mehod

Alhogh man hisorians of mahemaics have saed ha Kmmer's heo-r grew o of his work on Ferma's lasheorem, a carefl sd of mmerswork an correspondence inicaes haermas las heorem was raher inci-enal Kmmers main aim was o nhe solion o anoher problem in hearea of higher arihmeic, or nmberheor he problem of he higher reciproci laws pose b Gass (he probem of he higher reciproci laws con-siss in generaizing o higher powes heamos qdraic reciproci law haGass poved for second powers ri, he qaraic reciproci law saesha if an are od prime inegershen hee is a simpe raion bewenhe answers o he qesions os ier om he sqare of an ineger b a

mipe of ?" an oes ier fromhe sqare o n ineger b a mlipeo ?" n 1847 mmers work on hehigher reciproci laws was si in is er-l sags b b 185 he ha achiv



16/20

great suess proving a genera theoremthat was the umination of his work innumer theory The traditiona viewthat Kummer was motivated y interestin Fermats ast theorem is not entireyfase however eause the theorem isosey reated to te proem of thehigher reiproity aws Gauss himsefathough he aways denied that he was

interested in Fermats ast theorem perse expressed the hope that from his re-suts onerning the higher reiproityaws he woud one day e ae to dedueFermats theorem easiy

gr rms

Kummers theory of 87 was parti-uary vauae beause it provided asuient ondition for an odd prime to be an exponent for whih ermatsast heorem is true n other words if anodd prime satises Kummers ondi-tion then xP + yP z has no soution

In modern terminoogy a prime tha sat-ises the ondition is aed reguar(Speiay a prime is reguar if andony if it does not eveny divide the numerator of any of the rst - 3 num-ers in the series of frations aed theBernoui numbers ) Athough reguari-ty is a suient ondition for Fermat'sast theorem it is not a neessary ondi-tion There are primes that are notregar for whih x yP z has eenproved impossie Of the primes sma-er than 00 a ut 37 59 and 67 arereguar

Kummer jumped to the onusionthat the set of reguar primes was in-nite ut he soon reaized he oud notprove it to e true Indeed no suse-quent eort has yieded a proof a-though on the asis of intuition and nu-meria evidene the assertion that terere innitey many reguar primes seemsto e as ertain as an unproved state-ment an e (Oddy enough it has eenproved that there are innitey many ir-reguar primes Aout 60 perent of theprimes within the present range of om-putation are regar and there are goodreasons to eieve the majority of a

primes are reguar Hene a the primenumers an e divided into two su-sets the reguar primes and the irreguarprimes where the set that is expeted toe arger annot e proved innite utits ompement an e proved innite )

In ater years Kummer estaihedother suient onditions for Fermatsast theorem that oered even moreprime numers inuding the irreguarprimes 37 59 and 67 Sine Kummerstime even more inusie suient onditions have een found some of themost inusive of them by H S Van

diver of the University of Texas Eventhe most inusive onditions howeverhae sti not been shown to appy to aninnite number of prime exponentsThus it is sti oneivae no matter

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how unikely tat Fermats last teo-em is true only for nite number oprime exonents and consequentl thatere is some ver large number M suchthat te teorem is false for al rimeexonents greater tan M

On e oter hand, e sucient con-itios for te teorem are now so inclusive as to cover all te prime num

bers tat ave ever been tsted In oterwords al te rimes witin te range ofomutation ave been roved to be exponents for wic Fermats ast teoremis true Te algoritms for ceckingeter rimes satisf te conditionsre quite simle, and over te ast 25ears intensive cecking as been doneon arge modern comuters notably byD H Lemer of e Universit of Caliornia at Berkeley, R G Sefridge of teUniversity of Florida, Wels Jonson ofBowdoin Colege and Samuel S Wagsta of te University of Illinois In te

eary 197 0s J onson's computations esabised ta Fermats las teorem wasre for al prime exponents smallean 30000 ore ecently Wagsta,orking wit soisticated tecniquesnd large compute a te Universif Ilinois, as pused the limit st125 000

Tese computations lso so that ounterexample o emats ast heoem, se o integers x y z n p ohic x yP zP ould onsist fumbers o immense tha he oul bea beyon he ange of hand omputaion or even o omputaion on he largest existing comuters, indee a eond te range of n computer hat isemotely conceivable If p is prime beond Wagstas limit, say p is about300000, ten it can be sown thatxP + P is imossible unless x y orz is divisibe by p Terefore must beeater tan 300000, a number ofat east a milion digits Ote resultssow tat a counterexamle wouldinvove even more outlandisly large

nmbersTus in a certain sense Ferma's lasteorem is emirically true If tere is asoution for n + y n ten te num-bers in it are so inconc eivably large tathuan beings wil never be able to ealwit te From a ilosopical andmatematical oint of view, oweverte size of te numbers as no bearingon te validity of Fermat's las teoremWen a matematician says a statements true for all numbers e oe s not meanhat i is true only for all te numbersnone has ever encountered or everill encounter On te contrary sinceermat's last teorem as not even beenpoved true o an innite number ofpime exponents, one migt say tat ahe work on te roblem as done notng more tan verify te teorem in a

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Scientific American Volume 239 issue 4 1978 [doi 10.1038%2Fscientificamerican1078-104] Edwards,...

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Transcript of Scientific American Volume 239 issue 4 1978 [doi 10.1038%2Fscientificamerican1078-104] Edwards,...