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{\bf $\stackrel{\mbox{MA330 }}{\mbox{ (Sathaye)}}$
\hfill Quiz 2 \hfill  Name:\rule{2in}{0.01in} }
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\large

 Use the Brahmagupta-Jayadeva-Bh\={a}skara method to find a
smallest solution to the equation

$$\mathbf{ x^2-61y^2 =\pm 1.}$$

Be sure to show your work.
I need to see two answers: 

(1) The factored form as discussed. 

(2) The integer form after multiplying the factors and cleaning out the common factors.

(3) {\bf Hint:} Do remember to use the two Brahmagupta tricks (when the extended norm reduces to $\pm 2$ or $\pm 4$.)as discussed in class. You will need 2 or 7 steps depending on how well you use the tricks.
\footnote{This is the problem communicated by Fermat, but already solved by Bhaskara. The continued fraction for $\sqrt{61}$ comes out to be $[7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4]$.
If you evaluate the usual convergents, then you have to use the first 22 steps!!
The answer is huge: $1766319049, 226153980)$.} 





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