Kuṭṭaka


Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by = c where x and y are unknown quantities and a, b, and c are known quantities with integer values. The algorithm was originally invented by the Indian astronomer-mathematician Āryabhaṭa and is described very briefly in his Āryabhaṭīya. Āryabhaṭa did not give the algorithm the name Kuṭṭaka, and his description of the method was mostly obscure and incomprehensible. It was Bhāskara I who gave a detailed description of the algorithm with several examples from astronomy in his Āryabhatiyabhāṣya, who gave the algorithm the name Kuṭṭaka. In Sanskrit, the word Kuṭṭaka means pulverization, and it indicates the nature of the algorithm. The algorithm in essence is a process where the coefficients in a given linear Diophantine equation are broken up into smaller numbers to get a linear Diophantine equation with smaller coefficients. In general, it is easy to find integer solutions of linear Diophantine equations with small coefficients. From a solution to the reduced equation, a solution to the original equation can be determined. Many Indian mathematicians after Aryabhaṭa have discussed the Kuṭṭaka method with variations and refinements. The Kuṭṭaka method was considered to be so important that the entire subject of algebra used to be called Kuṭṭaka-ganita or simply Kuṭṭaka. Sometimes the subject of solving linear Diophantine equations is also called Kuṭṭaka.
In literature, there are several other names for the Kuṭṭaka algorithm like Kuṭṭa, Kuṭṭakāra and Kuṭṭikāra. There is also a treatise devoted exclusively to a discussion of Kuṭṭaka. Such specialized treatises are very rare in the mathematical literature of ancient India. The treatise written in Sanskrit is titled Kuṭṭākāra Śirōmaṇi and is authored by one Devaraja.
The Kuṭṭaka algorithm has much similarity with and can be considered as a precursor of the modern day Extended Euclidean algorithm. The latter algorithm is a procedure for finding integers x and y satisfying the condition ax + by = gcd.

Aryabhaṭa's formulation of the problem

The problem that can supposedly be solved by the Kuṭṭaka method was not formulated by Aryabhaṭa as a problem of solving the linear Diophantine equation. Aryabhaṭa considered the following problems all of which are equivalent to the problem of solving the linear Diophantine equation:
Aryabhata and other Indian writers had noted the following property of the linear Diophantine equations: "The linear Diophantine equation ax + by = c has a solution if and only if gcd is a divisor of c." So the first stage in the pulverization process is to cancel out the common factor gcd from a, b and c, and obtain an equation with smaller coefficients in which the coefficients of x and y are relatively prime.
For example, Bhāskara I observes: "The dividend and the divisor shall 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."

Aryabhata's algorithm

Aryabhata gave the algorithm for solving the linear Diophantine equation in verses 32–33 of Ganitapada of Aryabhatiya. Taking Bhāskara I's explanation of these verses also into consideration, Bibhutibbhushan Datta has given the following translation of these verses:
Some comments are in order.

Problem statement

Consider the following problem:

Data

Remainders = 15, 19
Greater remainder = 19
Divisor corresponding to greater remainder = 45
Smaller remainder = 15
Divisor corresponding to smaller remainder = 29
Difference of remainders = 19 - 15 = 4

Step 1: Mutual divisions

Divide 45 by 29 to get quotient 1 and remainder 16: 29 ) 45 29 16 13 3 ( 3
3
----
The process of mutual division stops here. 0

Step 2: Choosing an optional integer

Quotients = 1, 1, 1, 4, 3
Number of quotients = 4

Choose an optional integer = 2
The last quotient = 3
Multiply the optional integer by last quotient = 2 × 3 = 6
Add the above product to difference of remainders = 6 + 4 = 10

Step 4: Computation of successive numbers

Write elements of 1st column : 1, 1, 4, 3, 2, 4
Compute elements of 2nd column : 1, 1, 4, 10, 2
Compute elements of 3rd column : 1, 1, 42, 10
Compute elements of 4th column : 1, 52, 42
Compute elements of 5th column : 94, 52

The computational procedure is shown below:

Quotient 1 : 1 1 1 1 94
↗
Quotient 2 : 1 1 1 52 52
↗
Quotient 3 : 4 4 42 42
↗
Quotient 4 : 3 10 10
↗
k : 2 2

Difference : 4
of remainders

Step 5: Computation of solution

The last number obtained = 94
The residue when 94 is divided by the divisor corresponding to smaller remainder = 7
Multiply this residue by the divisor corresponding to larger remainder = 7 × 45 = 315
Add the larger remainder = 315 + 19 = 334

Solution

The required number is 334.

Verification of solution

334 = 11 × 29 + 15. So, 334 leaves a remainder of 15 when divided by 29.
334 = 7 × 45 + 19. So, 334 leaves a remainder of 19 when divided by 45.

Remarks

The number 334 is the smallest integer which leaves remainders 15 and 19 when divided by 29 and 45 respectively.

An example from ''Laghubhāskarīya''

The following example taken from Laghubhāskarīya of Bhāskara I illustrates how the Kuttaka algorithm was used in the astronomical calculations in India.

Problem statement

The sum, the difference and the product increased by unity, of the residues of the revolutions of Saturn and Mars – each is a perfect square. Taking the equations furnished by the above and applying the methods of such quadratics obtain the solution by the substitution of 2, 3, etc. successively. Then calculate the ahargana and the revolutions performed by Saturn and Mars in that time together with the number of solar years elapsed.

Some background information

In the Indian astronomical tradition, a Yuga is a period consisting of 1,577,917,500 civil days. Saturn makes 146,564 revolutions and Mars makes 229,6824 revolutions in a Yuga. So Saturn makes 146,564/1,577,917,500 = 36,641/394,479,375 revolutions in a day. By saying that the residue of the revolution of Saturn is x, what is meant is that the fractional number of revolutions is x/394,479,375. Similarly, Mars makes 229,6824/1,577,917,500 = 190,412/131,493,125 revolutions in a day. By saying that the residue of the revolution of Mars is y, what is meant is that the fractional number of revolutions is y/131,493,125.

Computation of the residues

Let x and y denote the residues of the revolutions of Saturn and Mars respectively satisfying the conditions stated in the problem. They must be such that each of. and is a perfect square.
Setting
one obtains
and so
For xy + 1 also to be a perfect square we must have
Thus the following general solution is obtained:
The value q = 2 yields the special solution x = 40, y = 24.

Computations of the ''aharganas'' and the numbers of revolutions

Ahargana is the number of days elapsed since the beginning of the Yuga.

Saturn

Let u be the value of the ahargana corresponding the residue 24 for Saturn. During u days, saturn would have completed ×u number of revolutions. Since there is a residue of 24, this number would include the fractional number 24/394,479,375 of revolutions also. Hence during the ahragana u, the number of revolutions completed would be
which would be an intger. Denoting this integer by v, the problem reduces to solving the following linear Diophantine equation:
Kuttaka may be applied to solve this equation. The smallest solution is

Mars

Let u be the value of the ahargana corresponding the residue 40 for Mars. During u days, Mars would have completed × u number of revolutions. Since there is a residue of 40, this number would include the fractional number 40/131,493,125 of revolutions also. Hence during the ahragana u, the number of revolutions completed would be
which would be an integer. Denoting this integer by v, the problem reduces to solving the following linear Diophantine equation:
Kuttaka may be applied to solve this equation. The smallest solution is