Indeterminate Equations-Aryabhata

Indeterminate Equations

A problem of great interest to Indian Mathematicians ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as Diophantine equation . This is an example from Bhaskara’s commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text sulbha sutras, whose more ancient parts might date to 800 BCE. Aryabhata’s method of solving such problems is called the kuṭṭaka (कुट्टक) method. Kuttaka means “pulverizing” or “breaking into small pieces”, and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm . The diophantine equations are of interest in cryptology, and the RSA Conference 2006, focused on the kuttaka method and earlier work in the Sulabhasutra.

 

Algebra

In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes:

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and

 

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(see squared triangular number)

 

to be continued……

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