Indeterminate Equations of the First Degree

Āryabhaṭa I (476) was the earliest Hindu Algebraist worked on the Indeterminate Equations of the First Degree. He provided a method for solving the simple indeterminate equation

$by-ax=c$

where a, b and c are integers.He also provided how to extend this to solve Simultaneous Indeterminate Equations of the first degree.

Bhāskara I (522) disciple of Āryabhaṭa I has displayed that the same method might be applied to solve the equation

$by-ax=-c$

and further that the solution of this equation would follow from that of

$by-ax=-1$

Brahmagupta and others followed the methods of Āryabhaṭa I and Bhāskara I

Importance

The subject of indeterminate analysis of the first degree was considered so important by ancient Hindu Algebraists that the whole science of algebra was once named after it. Āryabhaṭa II , Bhāskara II and others mentions precisely along with the sciences of arithmetic, algebra and astronomy.

On account of its special importance exclusive work on this entitled kuṭṭākāra śiromaṇi by Devarāja a commentator of of Āryabhaṭa I.

Varieties of Problems

There are three varieties of problems pertaining to Indeterminate equations of the first degree.

Variety 1 : Find a number (N) which when divided by two given numbers (a, b) will leave two remainders (R₁,R₂).

Now we have $N=ax+R_{1}=by+R_{2}$

$by-ax=R_{1}-R_{2}$

Putting $c=R_{1}\thicksim R_{2}$

$by-ax=\pm c$

the upper or lower sign being taken according as Rl

> or < R2• In a pro blem of the second kind we are

required to find a number (x) such that its product with

a given number (a) being increased or decreased .by

another given number .(v) and then divided by a 'third

given number (~) will leave no remainder. In other

words we shall have to solve

ax±- v =Y

~

in positive integers. The third variety of problems

similarly leads to equations of the form

by + ax = ± c.