10 V. S . VARADARAJA N

As mentione d above , Eule r (1707-1783) solve d th e Ferma t proble m (* ) fo r 7 2 = 3 . I t wa s i n th e

course o f hi s unsuccessfu l attemp t t o solv e (* ) fo r genera l 7 2 that Kumme r (1810-1893lai ) d th e

foundations o f moder n algebrai c numbe r theory .

1. Sho w tha t fo r th e equatio n bx + 2y = 2 5 th e solution s give n ar e th e onl y ones .

2. Chec k b y actua l calculatio n tha t ther e ar e n o solution s t o th e equatio n

X + y = Z (x,y, Z ar e positiv e integer s )

with x 5 , y 6 .

3. Th e Babylonian s kne w pythagorea n triplet s o f hug e sizes . Chec k tha t th e followin g ar e

pythagorean triplets :

z = 120,y= 119,^ = 169

x = 3456, y = 3367,z= 4825