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You are not logged in. #1 20090722 06:29:07
Pell’s equationToday, I read about Pell’s equation in the chapter on continued fractions of A Course in Number Theory by H.E. Rose. This is a Diophantine equation of the form where and is a fixed positive integer which is not a perfect square. Note that are always solutions to any Pell’s equation (called the trivial solutions). It can be proved that Pell’s equation always has a nontrivial solution (i.e. for which ) for all positive nonsquare integers . Last edited by JaneFairfax (20090722 08:13:27) #2 20090722 06:37:27
Re: Pell’s equationIs there an easy way of finding the nontrivial solutions? Why did the vector cross the road? It wanted to be normal. #3 20090722 08:10:18
Re: Pell’s equationHi mathsyperson and Jane; You start by computing the continued fraction of √14 The sequence is periodic with length 4 (1,2,1,6...) The nice part is that a theorem by Lagrange assures us that every square root like this will always have a repeating form. Compute the convergents of the √14. These are done by 2 recurence formulae or matrix multiplication You pick the 4th one in the sequence 15/4 and that is the smallest non trivial answer. This example is simple enough to get using this theorem. For any positive integer d, if d+2 is a perfect square, then d+1 is the first solution to Pell's Equation for x. I haven't seen a proof to this, just some web page. For harold the saxon problem: Compute the continued fraction of √61 The sequence is periodic with length 11 (1,4,3,1,2,2,1,3,4,1,14...) Compute the convergents of the continued fraction: So we pick the 11 term which is 29718/3805 but which is incorrect. So we try the next 11th (22nd term) term which is So this is the smallest solution that is not trivial. Here is a page to solve these and many tougher types of diophantine equtions. http://www.alpertron.com.ar/QUAD.HTM Here are other methods: This one uses matrices, it like the continued fraction approach is excellent for computers. http://fermatslasttheorem.blogspot.com/ … ution.html The most famous pell equation is the cattle of the sun. Last edited by bobbym (20090726 03:24:11) 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. #4 20090726 01:41:58
Re: Pell’s equationHere is an application of Pell’s equation in solving a numbertheory problem: Last edited by JaneFairfax (20090726 01:52:58) #5 20090727 17:43:37
Re: Pell’s equationHi Jane; one of the roots is phi ≈ 1.61803... Now just replace the x in the fraction with the entire statement with 1). Now again, replace x with 1). Again. You can truncate this at anytime to get the approximation. And their is an algorithm for higher order polys too. Last edited by bobbym (20090727 18:08:34) 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. #6 20090727 23:15:10
Re: Pell’s equationThanks. Rose (A Course in Number Theory) mentions various approximation techniques using continued fractions; for example, one method shows that is the first “best” appproximation to π. The next best approximation is Last edited by JaneFairfax (20090727 23:15:41) #7 20090728 06:21:32
Re: Pell’s equationHi Jane; Last edited by bobbym (20090728 06:22:22) 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. 