8 V. S. VARADARAJA N

How many way s there ar e t o divid e a flock of 2 5 sheep int o group s o f 5's an d 2's ?

Find tw o numbers suc h that th e square of either adde d t o the other give s a square .

Such problem s ar e know n a s diophantine problems an d ther e hav e bee n man y

famous one s i n history , suc h a s

Which number s ca n b e written a s a su m o f two squares ?

Can the sum of two cubes be a cube, a sum of two fourth power s be a fourth power ,

and s o on ?

The secon d o f these i s the famou s Fermat' s Las t Theore m whic h wa s posed b y

Fermat. Ferma t worked with a copy of Diophantus's Arithmetika edited by Bache t

and ha d commente d tha t

I hav e a truly marvellou s proo f tha t thi s canno t b e so, but ther e i s too littl e spac e

in th e margi n fo r m e to giv e it here .

Entire theorie s o f mathematic s wer e constructe d i n th e effor t t o solv e thi s

question. Indeed , one may say that moder n number theor y was born i n an attemp t

to devis e tool s fo r th e solutio n o f thi s problem . Th e conventiona l wisdo m i s tha t

Fermat wa s i n erro r an d di d no t posses s a proo f i n th e genera l case , althoug h h e

had a perfectl y legitimat e proo f fo r th e cas e n = 4 , whil e Eule r prove d th e resul t

for n = 3 much later . Th e proble m wa s unsolve d til l 1994 when Andre w Wiles ,

an Englis h mathematicia n workin g i n Princeto n University , stunne d th e worl d o f

mathematics b y finding a proof. Hi s proof o f course is not fro m first principle s an d

uses very sophisticate d idea s fro m moder n numbe r theor y an d algebrai c geometry ,

and i s published i n Annals of Mathematics, Vo l 141(1995), pp. 443-572 .

We shal l se e late r tha t i t wa s onl y afte r th e sixteent h centur y o r s o tha t al -

gebraic notatio n a s we now kno w slowl y cam e int o being . Howeve r on e can tr y t o

rewrite th e problem s mentione d abov e i n terms o f equations. Thu s th e proble m o f

dividing the 2 5 sheep into groups of 5's an d 2' s becomes the following. Le t u s write

x fo r th e numbe r o f groups o f 5 sheep an d y fo r th e numbe r o f groups o f 2 sheep.

Then w e must hav e th e equatio n

5x -f 2y = 2 5 (# , y ar e integer s 0 )

This equation ha s man y solution s which ca n be foun d ou t b y trial an d error . Thu s

we have

x = 1, y — 10, x = 3 , y = 5