Indeterminate Equations of the First Degree

In algebra, an indeterminate equation is an equation for which there is more than one solution.

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" the sanskrit names kṣepa, kṣepaka  etc. α is called the 'dividend' (bhājya), the unknown quantity (x) the "multiplier"  the sanskrit names guṇaka, guṇākāra etc.and y the "quotient" the sankrit name phala. The unknown (x) sometimes known by rāśi meaning "the unknown number" as per Mahāvīra.

Origin of the name:

The Sanskrit words kuṭṭa, kuṭṭaka, kuṭṭākāra, kuṭṭīkāra are all derived from the root kuṭṭ means "to crush", "to grind," "to pulverise". They all mean the act or process of "breaking", "grinding", "pulverising" as well as an instrument for that, that is, "grinder", "pulveriser".

Gaṇeśa(1545) says: "Kuṭṭaka is a term for the multiplier, for multiplication is admittedly called by words importing 'injuring,' 'killing.' A certain given number being multiplied by another (unknown quantity), added or subtracted by a given interpolator and then divided by a given divisor leaves no remainder; that multiplier is the Kuṭṭaka. so it has been said by the ancients. This is a special technical term."

Hence the subject of the indeterminate analysis of the first degree came to be designated by the term Kuṭṭaka.

Hindu method of solving the equation $$by-ax= \pm c$$ is based on a process of deriving from it successively other similar equations in which the values of the coefficients a,b become smaller and smaller. Hence the process is the same as that of breaking a whole thing into smaller pieces. Hence the ancient mathematicians adopted the name Kuṭṭaka for the operation.

Preliminary operations
In order that an equation of the form $$by-ax= \pm c$$ or $$by+ax=\pm c$$ may be solvable the two number a, b must not have a common divisor otherwise the equation would be absurd, unless the number c had the same common divisor. Hence the rule is a, b and c must be prime (dṛḍha = firm, niccheda = having no divisor,  nirapavarta = irreducible) to each other.

Bhaskara I observes: "The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.". Brahmagupta says: "Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other."

Hindu methods for the solution in positive integers of the equation $$by+ax=\pm c$$ we shall always take, unless otherwise stated a, b prime to each other.

Solution of $$by-ax= \pm c$$
Āryabhaṭa I's Rule:

Āryabhaṭa's problem is : To find a number (N) which when divided by two given numbers (a,b) will given R1 and R2

$$N=ax+R_1 = by+R_2$$

Denoting by c the difference between R1 and R2.

$$by=ax+c$$ if $$ R_1>R_2$$

$$ax=by+c$$ if $$R_2 > R_1$$

Hence $${\displaystyle x = {\frac{by+c}{a}}} \quad  {\displaystyle y = {\frac{ax+c}{b}}}  $$  according  as $$ R_1>R_2$$ or $$R_2 > R_1$$ a positive integer. Āryabhaṭa says : "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be multiplied by such an optional

Here a = divisor corresponding to greater remainder ; b= divisor corresponding to lesser remainder ; R1 = greater remainder ;                R2 = lesser remainder

Case - 1: if $$ R_1>R_2$$ then the equation to be solved will be $$by=ax+c$$  a, b being prime to each other.

"Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided (until the remainder becomes zero), the last quotient should be multiplied by an optional integer and then added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders. (Place the other quotients of the mutual division successively one below the other in a column; below them the result just obtained and underneath it the optional integer). Any number below (i.e., the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors."