Modern Mathematics in old Sanskrit books

Module 1The equation 61X2+1=Y2

Some Characters in this Story

• Pierre de Fermat, Germany• Frenicle, France• Euler, Switzerland • Lagrange, Italy• Brahmagupta, India• Bhaskara, India • Pell, England

Fermat to Frenicle :Translated by Jason Ross (Feb 1657)

what is the smallest square, which multiplied by 61 andadding 1, makes a square?What is the smallest square, which multiplied by 109 and adding

1, makes a square?If you do not send the general solution, send me the particular

answers of the two numbers that I have chosen (if you do not want too much work).

After having received your response, I will pose another question to you. It seems, without saying it, that my proposition is only for finding whole numbers which satisfy the equation, for, in the case of fractions, the least of arithmeticians would quickly come to an answer

Quote 1

• There is a sense of satisfaction, of fulfillment, in successful thinking……It’s notable too that the pleasure is in the solving of the problem. - See more at: http://perfectscoreproject.com/2011/06/why-kids-dont-like-school/#sthash.Ub4Tewdo.dpuf

According to Willingham, solving problems brings pleasure.

Quote 2

• Scientists and inventors have written of the sense of excitement that comes as they suddenly find the solution to a problem that has been on their minds for a long time. This excitement, they report, is one of life's highest pleasures.

• from Solve It! by James F. Fixx

What happened to that problem?

• Nobody could solve it during Fermat’s life-time. Later it was taken as a challenge by many bigwigs.

• Later Euler found an integer solution for 61X2+1=Y2.

• Still later, Lagrange gave a general method.

• They were not knowing that the problem was solved in India six hundred years before it was posed by Fermat.

Two great mathematicians

• French mathematician Fermat

• 1605 – 1665

• Leonhard Euler was a pioneering Swiss mathematician and physicist. (1707-1783)

Who solved it first?

• A general algorithm for solving all equations of this kind (in integers) was given by Bhaskaracharya II in his book Bijaganitam in the 11th century.

The shloka in which this problem is posed

• Which square number when multiplied by 67 and added to 1 will again be a square?

• Which again is a square number that when multiplied by 61 and added with 1 will yield a square root? (This second question is our today’s topic).

• का� सप्तषष्टिगु�णि त� का� तित रे�कायु�क्ता�

• का� चै�का षष्टिगु�णि त� चै सखे� सरूपा� |

• स्यु�त� मू�लदा� युदिदा का� तित: प्रका� तित: ति!त�न्त#

• तच्चे�तसिस प्रवदा त�त तत� लत�वत� ||

Some technical terms

• Kruti • Gunitaa• Yukthaa• Moolam • Roopam • Saroopaa • Mooladaa

• Square• Multiplied• Added• Square root • One• Together with 1.• Yielding a square root

Attack by trial and error method

• When X=1, 61X2+1 is 62. This is not a square.

• When X=2, 61X2+1 is 245. This is also not a perfect square.

• When X =3, …• If we proceed like this,

sooner or later, shall we not hit at a solution?

Though the existence of an integer solution is guaranteed, this method is not at all advisable.

Reason: One full span of human life is not enough to arrive at a solution by this method.

The least solution by Bhaskara’s algorithm

• y=1766319049, x=226153980

• We do not have 22 crores of minutes in 100 years.

• The fact: The same answer was obtained by Euler in 18th century. But in India, the general method was taught to students (in a textbook), since 1200 A.D.

• The verse where some terminology needed for this chakravala algorithm is introduced:

इ# ह्रस्व# तस्युवगु)* प्रका� त्यु�क्षु�ण् .यु�क्ता.वर्जि01त.व� स यु�!मू�ल# दाद्या�त� क्षु�पाका# त# ध! 4मू�ल#तच्चे ज्यु�ष्ठमू�ल# वदान्तिन्त

Two questions

• Do the foreign scholars give this credit to India?

• Yes, they do unequivocally.

• Is it an important discovery in mathematics?

• Yes, very much.

Four among many admirers of Bhaskara

• Hermann Hankel • Anglin • Cannor • Sir Thomas Little Heath

• German Mathematician• American Historian• Scottish Professor• British writer

Quote 1 of appreciation

• "It (Cyclic method of Bhaskara) is beyond all praise. It is certainly the finest thing achieved in the theory of numbers before Lagrange".

-- Hermann Hankel, (1839-1873) a German mathematician

remembered through Hankel transforms.

Quote 2 of appreciation

• "In many ways, Bhaskaracharya represents a peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations which is not to be achieved in Europe for several centuries".

• Robertson is a Professor emeritus of pure mathematics at the University of St Andrews. He is the author or co-author of seventeen textbooks

• Cannor is his co-worker in the website, Mac Tutor History of Mathematics.

Quote 3 of appreciation

• ''One of Bhaskara's feats in Number Theory consisted in finding the smallest positive integer solution of 61X^2+1=y^2 namely x= 226153980 and y=1766319049.“

• Year 1994.

Quote 4 of appreciation

• "Indian Cyclic Method of solving the equation x^2 - Ny^2 = 1 in integers due to Bhaskara in 1150 is remarkably enough, the same as that which was rediscovered and expounded by Lagrange in 1768".

• -Sir Thomas Little Heath

• FRS, mathematician, civil servant, historian, translator and mountaineer.

• 1861-1940.

Some keywords in the above quotes

• Beyond all praise.• Certainly the finest• Peak of knowledge• Remarkably enough• One of the feats• Achieved, not to be achieved in Europe

Quote

• ''Equations are a vital part of our culture. The stories behind them are fascinating.”

• One story is:Fermat aFrenchman wrote to Frenicle, his Friend

• Ian Stewart • Warwick University• In pursuit of the

unknown: 17 equations that changed the world

• Book published in 2012.

Quote

• "Coincidence is God's way of remaining anonymous".

• -Albert Einstein

• Otherwise why should Bhaskara and Fermat think of the same equation 61X2+1=Y2

With the same coefficient 61?

There is a mathematical explanation to this rare coincidence.

Two more great mathematicians

Lagrange• Used continued fractions to

arrive at general solutions.• Here the C.F.of square root

of 61 helps to obtain several solutions.

• Incidentally, the ideas behind continued fractions were first developed in India, centuries before Bhaskara.

Srinivasa Ramanujan• Was an expert in both Pell’s

equation and continued fraction.

• We shall soon see a problem of his.

Srinivasa Ramanujan(1887-1920)

Lived in Kumbakonam

Contributed to Number theory and combinatorics

Lagrange

• 1736-1813• Italy and France• He made significant

contributions to the fields of analysis, number theory, and mechanics.

A name to this family of equations

• An equation of the form NX2+1= Y2 needed a special name because it arose repeatedly in many applications. Euler coined the name “Pell’s equation”. It is a misnomer. The English mathematician John Pell did not contribute anything to this equation.

• It was Brahmagupta (7th century) who started obtaining important results about these equations.

• Bhaskara completely solved it.

• So a more suitable name would be Brahmagupta-Bhaskara equation.

• But it is difficult to change the name now.

Ramanujan’s Door Number problem

• In a certain street, there are more than fifty but less than five hundred houses in a row, numbered from 1, 2, 3 etc. consecutively. There is a house in the street, the sum of all the house numbers on the left side of which is equal to the sum of all house numbers on its right side. Find the number of this house.

• Answer is : 204th house among 288 houses.

• The solution involves the equation NX2+1= Y2

Bhaskara

(1114 – 1185) Indian

References to study further

• T.S.Bhanumurthy, A modern Introduction to Ancient Indian Mathematics, Wiley Restern (1992) (This book explains the mathematics of Bhaskara’s algorithm).

• S.P.Arya, UGC Jour. Mathematics Education (1980+). (This article also explains the Chakravala method).