This is a part of the series of lectures by Prof V Kannan from University of Hyderabad

1. Modern Mathematics in old
Sanskrit books
Module 1
The equation 61X2+1=Y2

2. Some Characters in this Story
• Pierre de Fermat, Germany
• Frenicle, France
• Euler, Switzerland
• Lagrange, Italy
• Brahmagupta, India
• Bhaskara, India
• Pell, England

3. Fermat to Frenicle :Translated by
Jason Ross (Feb 1657)
what is the smallest square, which multiplied by 61 and
adding 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

4. 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.

5. 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

6. 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.

7. Two great mathematicians
• French mathematician
Fermat
• 1605 – 1665
• Leonhard Euler was a
pioneering Swiss
mathematician and
physicist. (1707-1783)

8. 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.

9. 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 सप्तषष्टिगुणिता
कृ तत रेकयुक्ता
• का चैक षष्टिगुणिता च
सखे सरूपा |
• स्यात ् मूलदा यदद कृ तत:
प्रकृ तत: तितान्तं
• तच्चेतसस प्रवद तात
तता लतावत् ||

10. Some technical terms
• Kruti
• Gunitaa
• Yukthaa
• Moolam
• Roopam
• Saroopaa
• Mooladaa
• Square
• Multiplied
• Added
• Square root
• One
• Together with 1.
• Yielding a square root

11. 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.

12. 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:
इटिं ह्रस्वं तस्यवगगः प्रकृ ्या
क्षुण्िोयुक्तोवष्जगतोवा स येि
मूलं दद्यात ् क्षेपकं तं नििणे
मूलंतच्च ज्येटठमूलं वदष्न्त

13. 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.

14. Four among many admirers of
Bhaskara
• Hermann Hankel
• Anglin
• Cannor
• Sir Thomas Little Heath
• German Mathematician
• American Historian
• Scottish Professor
• British writer

15. 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.

16. 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.

17. 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.

18. 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.

19. 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

20. Quote
• ''Equations are a vital
part of our culture. The
stories behind them are
fascinating.”
• One story is:
Fermat a
Frenchman wrote to
Frenicle, his
Friend
• Ian Stewart
• Warwick University
• In pursuit of the
unknown: 17 equations
that changed the world
• Book published in 2012.

21. 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.

22. 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.

23. Srinivasa Ramanujan(1887-1920)
Lived in Kumbakonam
Contributed to Number
theory and
combinatorics

24. Lagrange
• 1736-1813
• Italy and France
• He made significant
contributions to
the fields of
analysis, number
theory, and
mechanics.

25. 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.

26. 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

27. Bhaskara
(1114 – 1185)
Indian

28. 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).