Pell's Equation and Bhaskara II
Arvind_Kolhatkar
An equation of the type
Nx^2+1=y^2 or
y^2-Nx^2=1 (where N, x and y are +ve integers)
is traditionally called Pell’s Equation. Bhaskar II, building on the
earlier work of Brahmagupta and others, has given in his BeejagaNita a
method called ‘Chakrawala’ for any value of N and has shown that for
N=61, y=1766319049 and x=226153980 is the smallest solution of the
equation. In other words
(1766319049)^2 - 61*(226153980)^2 = 1
Anyone can verify this by using the Calculator available in the
Accessories menu of the computer. It was also known that once a
solution is found for a Pell’s Equation, infinitely many more
solutions can be found by the iterative process.
This work done in India was not known to the Europeans. The so-called
Pell’s Equation makes its first appearance in Europe in 1657 when
Fermat (of the Last Theorem fame) challenged his contemporary
mathematicians in Europe and England to solve the very problem,
y^2-61x^2=1 that Bhaskara had mentioned in BeejagaNita, though even
the existence of the earlier work done in India was unknown in
Europe. How the very problem noticed and solved by Bhaskaracharya
reached to Fermat is not known. European mathematicians were able to
develop other methods to solve the problem.
Pell, an otherwise little known mathematician, has been immortalized
by having his name attached to the equation, which honour, we now
know, rightfully belongs to Bhaskaracharya. Pell got his name
attached to the equation because Euler, another giant of European
mathematics, called it so, mistakenly believing that the work was
Pell’s when it was really done by another well-known mathematician,
Brouncker.
For a more detailed discussion of the history of this problem and of
the older Indian contribution, please visit
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html
Arvind Kolhatkar, March 17, 2011.
murthy
equation is well researched and accepted by mathematicians, I believe. The
book by Dr. Varughese -I may be excused if I have not spelled his name
correctly- title of which I forget (Word "peacock" occurs in the title if I
remember it right) dicusses this and there is much info on web too.
As usual giving credit through naming has inherent "social drags".
Murthy
----- Original Message -----
From: "Arvind_Kolhatkar" <kolhat...@gmail.com>
To: "samskrita" <sams...@googlegroups.com>
Sent: Friday, March 18, 2011 2:10 AM
Subject: [Samskrita] Pell's Equation and Bhaskara II
Dear Group,
An equation of the type
Nx^2+1=y^2 or
y^2-Nx^2=1 (where N, x and y are +ve integers)
is traditionally called Pell�s Equation. Bhaskar II, building on the
earlier work of Brahmagupta and others, has given in his BeejagaNita a
method called �Chakrawala� for any value of N and has shown that for
N=61, y=1766319049 and x=226153980 is the smallest solution of the
equation. In other words
(1766319049)^2 - 61*(226153980)^2 = 1
Anyone can verify this by using the Calculator available in the
Accessories menu of the computer. It was also known that once a
solution is found for a Pell�s Equation, infinitely many more
solutions can be found by the iterative process.
This work done in India was not known to the Europeans. The so-called
Pell�s Equation makes its first appearance in Europe in 1657 when
Fermat (of the Last Theorem fame) challenged his contemporary
mathematicians in Europe and England to solve the very problem,
y^2-61x^2=1 that Bhaskara had mentioned in BeejagaNita, though even
the existence of the earlier work done in India was unknown in
Europe. How the very problem noticed and solved by Bhaskaracharya
reached to Fermat is not known. European mathematicians were able to
develop other methods to solve the problem.
Pell, an otherwise little known mathematician, has been immortalized
by having his name attached to the equation, which honour, we now
know, rightfully belongs to Bhaskaracharya. Pell got his name
attached to the equation because Euler, another giant of European
mathematics, called it so, mistakenly believing that the work was
Pell�s when it was really done by another well-known mathematician,
Brouncker.
For a more detailed discussion of the history of this problem and of
the older Indian contribution, please visit
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html
Arvind Kolhatkar, March 17, 2011.
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Vis Tekumalla
May be it's not "social drag." Could be honest unawareness of earlier work. For example, the formula for the area of cyclic quadrilateral (quadrilateral inscribed in a circle) is known to this day as Brahmagupata's formula. The formula is -
cyclic quad area = sqrt ((s-a)(s-b)(s-c)(s-d)), where s = semiperimeter, and a,b,c,d are the sides.
...Vis Tekumalla --- On Fri, 3/18/11, murthy <murt...@gmail.com> wrote: |
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murthy
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