By V. S. Varadarajan

This article deals a unique account of Indian paintings in diophantine equations through the sixth via twelfth centuries and Italian paintings on suggestions of cubic and biquadratic equations from the eleventh via sixteenth centuries. the quantity strains the historic improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few sleek topics akin to the elemental theorem of algebra, Clifford algebras, and quarternions. it truly is aimed toward undergraduates who've no heritage in calculus.

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Pk,Qk) = 1 . Henc e verif y tha t (pk-> qk) (2pfc_ i - f - 3 ^ - i , P f c - i + 2^-i ) Show tha t th e first si x o f thes e solution s ar e (2,1), (7,4) , (26,15), (97,56) , (362,209) , (1 351 ,780 ) Also startin g wit h th e solutio n ( 1 , 1) t o X — 3Y = — 2 verif y tha t (1,1), (5,3)(1 9,1 1 ) , (71 ,41 ) , (265,1 53 ) are als o solution s t o th e sam e equation . Henc e deduc e th e approximation s t o 2 6 5 r- 1 35 1 I53

Halve th e numbe r o f roots , whic h i n thi s cas e yield s five . Thi s yo u multipl y by itself ; th e produc t i s twenty-five. Ad d thi s t o thirty-nine ; th e su m i s sixty-four . Now tak e th e roo t o f this , whic h i s eight , an d subtrac t fro m i t hal f th e numbe r o f roots, whic h i s five. Th e remainde r i s three. Thi s i s the roo t yo u aske d for . This se t o f instruction s applie s t o th e equatio n X2 + 2BX = C giving th e sequenc e o f step s X2 + 2BX + B 2 = C + B 2 (X + B ) 2 = C + B 2 X + B = VC + B2 X = y/C + B 2 - B There i s n o nee d t o dwel l to o muc h o n ho w cumbersom e wer e th e processe s available t o peopl e i n thos e day s i n discoverin g an d communicatin g mathematica l ideas an d calculations .

I f & is a positive intege r sho w tha t (2A : + 1 ) = 4k + 4k + 1 and deduc e tha t th e squar e of a n od d intege r i s not only od d bu t leave s th e remainde r 1 whe n divide d b y 4. Usin g a similar argument , sho w tha t th e squar e of an intege r no t divisibl e b y 3 leaves th e remainde r 1 whe n divide d b y 3. For provin g tha t th e description give n abov e o f all primitive triplet s wit h a eve n i s correct one needs to know tha t ever y intege r i s a product o f primes (repetition s allowed ) an d tha t thi s ca n be done i n a unique manne r if we disregard th e orde r i n which th e primes appea r i n the factorization .

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