New Correction Page

[Artur]

Dear Neil,
About new http://www.research.att.com/~njas/sequences/submitC
I don't understand reason why is giving two times oryginal sequence 
(after preview) and all corrections are in as narrow window.

After preview in my opinion should be oryginal one (only one time) and 
all corrections visible in expanded Window with unlimited length.

Best wishes
Artur





Any equation of the form ax^2-by^2 = c with a,b,c integers, a,b>0 and ab 
not a square can be converted to a generalized Pell equation u^2-abv^2 = ac 
of the form u^2-Dv^2=n. This can be solved using standard methods (e.g. 
compute the convergents of the continued fraction expansion of sqrt(D)). 
The solutions (x,y) of the original equation are in 1-1 correspondence to 
those solutions (u,v) of the generalized Pell equation that additionally 

In your particular example, the generalized Pell equation is u^2-12v^2=4, 
which has two infinite families of (positive) solutions:


Where k is any nonnegative integer. The first family has u divisible by 4 
and yields an infinite set of solutions to your original equation 
(x,y)=(u/4,v), whose first three terms (n=0,1,2) correspond to the 
divisible by 4. It follows from standard results on the generalized Pell 
equation (e.g. Thm 5.13 of Mollin's "Fundamental Number Theory with 
Applications) that this analysis accounts for all solutions over the 
natural numbers.

Drew


On Sep 4 2008, c.zizka at email.cz wrote:

>Dear Seqfans,
>
>Root Mean Square (1, 3, ... , 2*K-1) = C   (C,K positive integers)
>
>gives a nice Pell-like equation :
>
>4*K*K - 3*C*C = 1
>
>Solutions :
>K = 1 , C = 1
>K = 13 , C = 15
>K = 181 , C = 209 
>
>Are there other solutions ?
>
>Thanks for comments,
>
>Ctibor
>