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You are not logged in. #26 20121018 23:38:12
Re: EquationSo, generally you require a calculator to solve a Pell equation...? #27 20121018 23:42:13
Re: EquationDepends on the Pell equation. For some of them like you can tell just by looking at them. There is the method of Brahmagupta and Bhaskara two ancient Indian mathematicians. Generally though, in computational number theory you will require the use of at least a good scientific calculator. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #28 20121019 00:01:56
Re: EquationYou can tell just by looking at them? You mean, finding the fundamental solution using their method, or by trial and error? #29 20121019 00:08:04
Re: EquationYes, once you have the fundamental solution ( or any solution other than the trivial one ) you use a recurrence to get more solutions.
By inspection x = 24 and y = 5. For that one you must compute the answer. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #30 20121019 01:24:44
Re: EquationI must compute the answer for that one? I don't understand... #31 20121019 01:31:42
Re: EquationWhen I say by inspection I mean without computing the convergents. The OP's one requires some computation. The other is as simple as x +2 = 4. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #35 20121019 01:49:06
Re: EquationYou used the recurrence to get that. But if you want to get the fundamental solution convergents of the continued fraction is best. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #36 20121019 01:51:41
Re: EquationBut how can I compute them by hand? These problems often come to me when I have no computer or calculator near me, such as in an olympiad, or if someone comes up to me with a problem like it. #37 20121019 01:57:10
Re: EquationThey never come up in an olympiad. The only time someone asks that is when they know nothing and think the problem is easy. Guys like that think everything is as easy as plugging into the quadratic formula. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #39 20121019 02:08:51
Re: EquationThey could not approach it. It is a problem that is historically famous. This is how the problem begins.
In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #40 20121019 02:18:45
Re: EquationHow do you form the Pell equation from that? #41 20121019 02:23:55
Re: EquationThe equation in post #37 is the Pell equation for that problem. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #42 20121019 02:30:32
Re: EquationThat is interesting, especially how they were only able to get the solution (all the digits) in 1965. #43 20121019 02:35:38
Re: EquationIt 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. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #44 20121020 00:34:04
Re: EquationThank you,now I can find a pell's equation's fundamental solution,but what is the method of finding the additional ones?please explain. #45 20121020 00:38:50
Re: EquationYou use two recurrences to find more answers. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #47 20121020 01:06:34
Re: EquationHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #48 20121020 01:29:07
Re: EquationWhat ways could you please explain.(sorry for disturbing you so much) #49 20121020 01:36:50
Re: EquationHi Leroy; With Running the recurrences in the forward direction we get the first bunch: In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #50 20121021 12:12:57
Re: EquationHi again,I have understood pell's equation,now I am curious about nagetive pell's equation,so is there any method to solve x^2ny^2=1 