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Forms and methods of solutions of indeterminate equations of orders greater than one by Ancient Indian mathematicians Vivek Sinha (09 MS 066) IISER Kolkata In this paper the forms and methods of solutions of indeterminate equations of orders greater than one by some eminent ancient Indian mathematicians like Bhaskaracharya (II), Brahmagupta, Nrayana Pandit etc. have been discussed with highlighting examples. Special focus has been given to the cyclic (chakravala) method for solving the Diophantine equation of second degree viz. Nx2 + K = y2. “Time and again world has looked up upon us such that we know nothing, and of course, we gave them nothing ( the zer0)!” Anonymous 1. INTRODUCTION Indeterminate equations of orders greater than one fall under the special category of equations called Diophantine equations. *Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. Examples of Diophantine equations* In the following Diophantine equations, x, y, and z are the unknowns, the other letters being given are constants. This is a linear Diophantine equation For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat’s last theorem states that no positive integer solutions (x, y, z) satisfying the equation exist. (Pell’s equations) which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century. * http://en.wikipedia.org/wiki/Diophantine_equation India's contribution to integral solutions of Diophantine equations can be traced back to the Sulbha Sutras, which were Indian mathematical texts written between 800 BC and 500 BC. Baudhayana (circa 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also attempted to solve simultaneous Diophantine equations with up to four unknowns. Apastamba(circa 600 BC) attempted simultaneous Diophantine equations with up to five unknowns. Diophantine equations were later extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for determination of integral solutions of Diophantine equations. Systematic methods for finding integer solutions of Diophantine equations could be found in Indian texts from the time of Aryabhata AD (499). Brahmagupta (628) handled more difficult Diophantine equations - he investigated Pell's equation, and in his Samasabhavana he laid out a procedure to solve Diophantine equations of the second order, such as 61x2 + 1 = y2. It is interesting to note that these methods were unknown in the west, and in fact this very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat. It was solved seventy years later by Euler. However, the solution to this equation was found more than centuries ago by Bhaskara II in his book Lilavati (1150) (who also found the solution to Pell's equation), using a modified version of Brahmagupta's method(Brahma Sphuta Siddhānta). Bhaskara II, in his Bijaganita (Algebra), recognized that a positive number has both a positive and negative square root, and solved quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations. It is also amazing to know that the Pell’s equation was never solved by Pell. In fact this equation should rightly be called as Brahamgupta’s equation. We shall now discuss the cyclic method of solving the indeterminate quadratic 2 2 equation, Nx + K = y . 2.CYCLIC METHOD FOR SOLUTION OF THE INDETERMINATE EQUATION, 2 2 Nx +k = y 2.1Brahmagupta’s lemma: 2 2 If (s,t) is a positive integral solution(found by trial) to the equation Nx +k = y 2 2 2 2 2 2 2 then (2st,Ns + t ) is a solution to Nx +k = y , or (2st/k,(Ns + t )/k) is a solution to Nx2+1 = y2. Here N, s, t >0 and are integers. Bhaskara’s improvement: The cyclic method improves the Brahmagupta’s method. Here, we don’t have to look for one solution by trial but we can get it by calculation. The proof of Brahamgupta’s lemma and Bhaskara II’s further work has been given in supplementary material. We shall now see an example on application of Brahmagupta’s Samāsa operation. EXAMPLE 1 Consider the equation 3x2+1 = y2. On comparing with the standard form of the equation Nx2+k = y2, we have N = 3 and k = 1. By observation, we find one integral solution (1,2). Applying Samāsa operation between the solution (1,2) with itself we get that (4,7) is also a solution. Now applying the same operation between (1,2) and (4,7) we get that (15,26) is also a solution. Continuing this way we get that(1,2),(4,7),(15,26),56,97),(209,362)…….. (infinite number of sets). Now we will see an example based on the Cakravāla method. EXAMPLE 2 Let us consider the equation 61x2+1= y2. (1,8) is an integral solution to the equation 61x2+k= y2 with k = 3. In the identity given in Bhaskara’s work, we choose m so that (m+8)/3 is an integer ams |m2-61| is least. We get m = 7, and for this value of m, the identity becomes 61*5*5 – 4 = 39*39. Thus (5/2,39/2) is a solution of the equation 61x2-1= y2, and this is in a standard form. Using Brahmagupta’s Samāsa operation , we find that (195/2,1523/2) is a solution of the equation 61x2+1= y2. Now applying Samāsa operation between (5/2,39/2) and (195/2,1523/2) we get (3805,29718). Repeating the process between (3805,29718) and itself gives (226153980,1766319049) as a solution. Narayan Pandit, another eminent ancient Indian mathematician is also known to work on indeterminate equation s of order higher than one. Following is an example taken from his book Bijganita Vatamsa written in 1350 A.D. EXAMPLE 3 Consider the equation 97x2+1 = y2. Here we see that (1,10) is an integral solution of 97x2+k = y2, with k = 3. In Bhaskar’s identity, mis to be so chosen that (m+10)/3 is an integer and |m2-97| is least. This gives m = 11, and for this value of m, the identity becomes 97*(7)2+8 = (69)2. Thus (7,69) is an integral solution of 97x2+8 = (69)2. For the latter equation, u = 7, v = 69,k= 8 and N = 67. m is now to be chosen such that (7m+69)/8 is an integer and |m2-97| is least, thus giving m = 13, and for this value of m, the identity we have is 97*(20)2+9 = (197)2. Thus (20,197) is an integral solution of 97x2+9=y2. Now for this equation u = 20,v = 197 and k = 9. So now we choose m such that (20m+197)/9 is an integer and that |m2-97| is least. We can see that m = 5. So the identity becomes 97*(33)2+8 = (325)2, showing that (33,325) is an integral solution of 97x2+3 = y2. For this equation u = 33, v= 325 and k = 8. So now we choose m in such a fashion that (33m+325) /8 is an integer and |m2- 97| is least. We have m = 11, and for this the identity that we have is 97*(86)2+3 = 8472. Thus the new equation that we have Is 97x2+3 = y2 with (86,847) as an integral solution. Now for this equation u = 86, v= 847 and k = 3. So now we choose m such that (86m+847)/3 is an integer and |m2-97| is least. We see that m = 10 satisfies such conditions. So the Bhaskara’s identity becomes 97x2+1 = y2. But this equation is in a standard form and its solution therefore can be obtained as [x,y] = [(2*569*5604)/1,(97*(569)2+(5604)2)/1] Thus x = 6377352,y = 62809633 by applying Samāsa method. 3. DISCUSSION If we examine the methods used to solve the indeterminate equations closely we can marvel at the genius of these classic mathematicians. One can see the reflection of an intuition guided marvellous and extraordinarily simple solution to apparently complex problems. The algorithm used for solving the Diophantine equations is something similar to what one uses in modern day programming. As we see in example no. 2, mere verification of the solution is not easy, even if one uses a normal calculator! No wonder that people using methods developed by ancient Indians for basic mathematical operations can calculate really fast (even faster than a computer some times!). Highly intellectual and given to abstract thinking as they were, one would expect the ancient Indians to excel in mathematics$.