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I took this 5 hour online course on the Aryabhata algorithm for finding integer solutions for
a x + b y = c with co-prime a,b.
It was enjoyable. The instructor's @bseshadri love for math was nice to experience in a mostly & sadly math phobic world.
More comments next. https://twitter.com/GenWise_/status/1270241795172007936
a x + b y = c with co-prime a,b.
It was enjoyable. The instructor's @bseshadri love for math was nice to experience in a mostly & sadly math phobic world.
More comments next. https://twitter.com/GenWise_/status/1270241795172007936
To begin with, though familiar with integer solutions of
a x + b y = c,
didn't know details of Aryabhatta algorithm. It is a clever reductive implementation (hence the name Kuttuka or breaking down) of expressing GCD/HCF by reversing Euclid's algorithm.
5th Century. Wow.
a x + b y = c,
didn't know details of Aryabhatta algorithm. It is a clever reductive implementation (hence the name Kuttuka or breaking down) of expressing GCD/HCF by reversing Euclid's algorithm.
5th Century. Wow.
Now Aryabhatta Kuttuka algorithm is essentially built on reversal of the reductive Euclidean algorithm for finding greatest common factor.
In some sense Kuttuka aspect was inherited from Euclid's algorithm itself!
But the constructive recipe valid for all cases is impressive!
In some sense Kuttuka aspect was inherited from Euclid's algorithm itself!
But the constructive recipe valid for all cases is impressive!
Natural question: Were the notions of GCD and Euclid algorithm known by that time? When did these become known?
No taking away from the brilliant and elegant algorithm, disappointing part is apparently no attempt at formalization over next few centuries. Compare that with ..
No taking away from the brilliant and elegant algorithm, disappointing part is apparently no attempt at formalization over next few centuries. Compare that with ..
So no formalization of the Kuttuka approach in the next few centuries, in contrast to, for example, formalization of zero and negative numbers hy Brahmagupta.
Also did the Indian mathematics schools miss the notion of prime numbers which were known to the Greeks? How come?
Also did the Indian mathematics schools miss the notion of prime numbers which were known to the Greeks? How come?
Sticking my neck out I would speculate that by and large Indian mathematics ignored formalization in preference to constructivist and computational approaches.
While I appreciate the ingenuity and elegance of algorithm development, I miss in it the beauty of Greek Geometry.
While I appreciate the ingenuity and elegance of algorithm development, I miss in it the beauty of Greek Geometry.
Net net: Aryabhatta in 5th Century nailed down in a few lines a constructive proof of Bezhuit identity derived around 1780.
This is amazing and stunning by any standard.
We should mention this algorithm in number theory books.
That will be a fitting tribute Aryabhatta.
Amen.
This is amazing and stunning by any standard.
We should mention this algorithm in number theory books.
That will be a fitting tribute Aryabhatta.
Amen.
