ALGEBRA I N ANCIEN T AN D MODER N TIME S 9
are solutions. Usuall y in algebra, if there are 2 unknown quantities to be solved for ,
there wil l b e tw o equations ; her e ther e i s onl y on e equation , an d w e ar e require d
to find al l solutions , bu t wit h th e additiona l requiremen t tha t x an d y b e bot h
positive integers . Thi s i s a n exampl e o f a diophantine equation. W e shal l se e i n
the nex t chapte r tha t mathematician s fro m Indi a i n the period fro m th e 6 th t o th e
12th centurie s mad e a deep stud y o f the diophantin e equation s
x2
- Ny
2
= ± 1
where N i s a positiv e intege r whic h i s no t a squar e an d (x , y ar e t o b e positiv e
integers.
Turning t o th e nex t se t o f equations , th e questio n o f whic h square s ar e th e
sum o f two squares, the equatio n i s
x2
- f y
2
= z
2
(x , y, z ar e integer s 0 )
Every on e is familiar wit h th e solution s
x 3 , y = 4 , z = 5 , x = 5 , y = 12, z = 13
Clearly, i f x an d y ar e th e length s o f th e side s o f a righ t triangl e tha t contai n th e
right angle , the n z i s th e lengt h o f th e hypotenuse . Becaus e o f thi s connectio n
with th e Pythagora s theorem , th e set s o f positiv e integer s (x , y, z) satisfyin g th e
equation
2 I 2 2
x -f y = z
are calle d pythagorean triplets. W e shal l loo k int o the m mor e closel y i n th e nex t
chapter. Th e ancient s kne w way s t o generat e thes e triplet s wher e x , t/, z ar e hug e
numbers.
N O T E S AN D EXERCISE S
Pierre d e Ferma t (1601-1665i ) s generall y regarde d a s th e founde r o f moder n Number
theory, a branc h o f mathematic s generall y dealin g wit h whol e number s an d rationa l numbers ,
and diophantin e equations . H e develope d systemati c method s fo r studyin g certai n classe s o f
diophantine equation s an d discovere d man y remarkabl e an d dee p propertie s o f thei r solutions .
The reade r wh o want s t o kno w mor e abou t th e histor y o f numbe r theor y an d th e me n suc h a s
Fermat wh o created it , should rea d th e wonderful boo k [W ] of Andre Wei l [(1906-1998)]on , e o f
the greates t mathematician s an d numbe r theorist s o f this century .
Fermat's problem is to show that fo r 71 3 ther e are no solutions i n positive integer s X, y, Z
of th e equatio n
x
n
+ y
n
= Z
n
(* )
Fermat prove d thi s fo r n = 4 b y showin g mor e generall y tha t ther e ar e n o solution s i n positiv e
integers X, ?/, Z of th e equatio n
x4 + y 2 = z 4
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