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Topic: Pell type equations
Replies: 29   Last Post: Apr 29, 2012 5:37 PM

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Dan Cass

Posts: 442
Registered: 12/6/04
Re: Pell type equations
Posted: Apr 26, 2012 2:54 PM
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> Hi,
>
> 5x^2 + 20 = y^2
>
> I can get valid integer values for x and y using
> brute force (1, 5), (4,
> 10), (11, 25) etc.
>
> I've been going through the work on this page...
>
> http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pel
> l.html
>
> which covers the work of Brahmagupta and Bhaskara II
> on Pell type equations.
> I can't seem to figure if this is any use for
> calculating integer roots for
> the above equation.
>
> Any help would be greatly appreciated.
>


Put your equation as y^2 - 5*x^2 = 20.
You have the initial solution y=5, x=11.

A solution to y^2 - 5*x^2 = 1 is y=9,x=4.
So a list of solutions to your eqn is obtained by
choosing some positive integer n and "multiplying out"
the expression (5 - sqrt(5))*(9-4*sqrt(5))^n.
Then put y=integer part and x=integer before the sqrt(5).

First of these is y=65, x=29.

Note there are more than the solutions obtained this way.



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