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- bobbym
- 2012-10-21 17:25:54
No, there is a method to find the convergents.
- Leroy
- 2012-10-21 17:18:42
So I have find the convergent by trial and error
- bobbym
- 2012-10-21 15:13:38
There is a theorem that says that. This is not something I just cooked up, I wished I had.
- Leroy
- 2012-10-21 14:17:31
I have understood everything quite well,but I don't understand what is the logic for taking the convergents,is there any rule for which convergent will be the solution?
- bobbym
- 2012-10-21 12:28:50
Hi Leroy;
Try this site first:
http://en.wikipedia.org/wiki/Pell%27s_e … l_equation
- Leroy
- 2012-10-21 12:12:57
Hi again,I have understood pell's equation,now I am curious about nagetive pell's equation,so is there any method to solve x^2-ny^2=-1
- bobbym
- 2012-10-20 01:36:50
Hi Leroy;
No bother at all. You remember that the fundamental solution is x = 10 and y = 3. We use that to get all the rest.
With
Running the recurrences in the forward direction we get the first bunch:
- Leroy
- 2012-10-20 01:29:07
What ways could you please explain.(sorry for disturbing you so much)
- bobbym
- 2012-10-20 01:06:34
Hi;
Yes, that is one way if you have the convergents. x=numerator and y=denominator of the period.
There are other ways that do not require the convergents.
- Leroy
- 2012-10-20 01:02:16
You mean every 2nd convergent after the fundamental one?
- bobbym
- 2012-10-20 00:38:50
You use two recurrences to find more answers.
- Leroy
- 2012-10-20 00:34:04
Thank you,now I can find a pell's equation's fundamental solution,but what is the method of finding the additional ones?please explain.
- bobbym
- 2012-10-19 02:35:38
It would seem so. Vardi and others believe he at least formulated the problem correctly even if he was unable to solve for the 205 000 digit answer.
- zetafunc.
- 2012-10-19 02:30:32
That is interesting, especially how they were only able to get the solution (all the digits) in 1965.
So, Archimedes probably knew that equations of that form were solvable.
- bobbym
- 2012-10-19 02:23:55
The equation in post #37 is the Pell equation for that problem.
Read this for more:
http://www.sciencenews.org/sn_arc98/4_1 … thland.htm
It took more that 1800 years for someone to partially solve the problem. It was only solved fully in the 20th century with the aid of computers.
Incidentally, the Pell equation is incorrectly named. Pell had nothing to do with it. It should be called Fermat's equation.