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.

Types of Problems

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

Type 1: Find a number N which when divided by two given numbers a and b will leave two remainders R₁ and R₂ .

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

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

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

$by-ax=\pm c$

Positive or Negative sign considered according as R₁ is greater than or less than R₂.

Type 2:

Find a number 'x' such that its product with a given number 'α' being increased or decreased by another given number 'γ' and then divided by a third

given number 'β' will leave no remainder. In other words we shall have to solve

${\frac {\alpha x\pm \gamma }{\beta }}=y$

in positive integers.

Type 3: Equations of the form ${\displaystyle by+ax=\pm c}$

Terminology

Hindus called the subject of indeterminate analysis of the first degree as kuṭṭaka , kuṭṭākāra, kuṭṭīkāra or simply kuṭṭa. The names kuṭṭākāra and kuṭṭa appear as early as the Mahā-Bhāskarīya of Bhāskara I (522) . In the commentary of Āryabhaṭīya by Bhāskara I the terms kuṭṭaka and kuṭṭākāra can be found. The terms kuṭṭaka , kuṭṭākāra and kuṭṭa was used by Brahmagupta. Mahāvīra had a preferential liking for the term kuṭṭīkāra.

In the type 1 problem the quantities a , b are called "divisors", the sanskrit names are bhāgahāra, bhājak, cheda etc.) and R₁ and R₂ "reminders", the sanskrit names are agra, śeṣa etc.

In the type 2 problem β is called the "divisor" and γ is called the "interpolator"