1. Show that k=15 is congruent integer.

Question

where 0.5xy=km^2,
x=r(u^2 - v^2)
y=r(2uv)
z=r(u^2 + v^2)
x^2 + y^2 = z^2

2. how i solve Pell's equation using bhaskara II then brahmagupta method for x^2 - 67y^2 =1

Please help, exam on monday... :)

Answer 1

#2
I don't know brahmagupta's method, but I remembered a method that uses continued fractions.
It is a rather easy method compared to some of the other methods that are out there
It involves finding the continued fraction terms of ?67
which are
8; 5,2,1,1,7,1,1,2,5,16
Then I complete a complicated chart and found
x = 48842 , y = 5967 to be a solution

(48842)^2 - 67(5967)^2 = 1

continued fractions are an amazing topic
and if you want to learn more here is start:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html

an remarkable page!!!

Answer 2

Yea i know... but my lecturer wants us to solve it using that method.... actually i know how to use brahmagupta method... but i just cant get bhaskara II to transform into something like brahmagupta