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Topic: An equation
Replies: 4   Last Post: Aug 19, 2011 11:05 AM

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dushya

Posts: 31
Registered: 11/18/09
Re: An equation
Posted: Aug 19, 2011 11:05 AM
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On Aug 19, 1:48 pm, "Philippe 92" <nos...@free.invalid> wrote:
> Tim Little wrote :
>
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>
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>
>
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>

> > On 2011-08-19, dushya <sehrawat.dushy...@gmail.com> wrote:
> >> On Aug 19, 5:55am, dushya <sehrawat.dushy...@gmail.com> wrote:
> >>> Does following equation has any "large" positive integer solutions for
> >>> N,M,P --

>
> >>> ((N^2)+N) ((M^2)+M) = 6 ((P^2)+P)
>
> >>> one family of solutions is, N=2, and M=P (or M=2, and N=P)
>
> >>> But I am looking for solutions in which P >3, and also one of N and M
> >>> is greater than 3.

>
> >> Correction -- I am looking for solutions in which BOTH N and M are
> >> greater than 2.

>
> > Sure, for example N=3 corresponds to triangular numbers that are twice
> > another triangular number, giving infinitely many solutions in M and
> > P.  E.g.http://oeis.org/A029549

>
> > There are arbitrarily large solutions.  Many of these can be found by
> > considering it in terms of the question "which natural numbers can be
> > written as the ratio of two triangular numbers?" answered e.g. at

>
> >  http://mathoverflow.net/questions/69277/ratio-of-triangular-numbers
>
> Also :
> Let N = 23 (why not)
> 23*24*(M^2 + M) = 6(P^2 + P)
> that is
> 92M^2 + 92M = P^2 + P
> (a Pell equation after a suitable variable substitution)
> plugging this into the Alpetronhttp://www.alpertron.com.ar/QUAD.HTM
> gives the infinitely many solutions :
>
> M = -1, P = 0 (results into no >0 solutions)
> and also: M = 0, P = -1 (--> (515, 4944) etc)
> and also: M = -1, P = -1  (results into no >0 solutions)
> and also: M = 2, P = 23 (all >0 solutions)
> and also: M = 2, P = -24 (--> (57, 551) etc.)
> and also: M = 57, P = 551 (all >0 solutions)
> and also: M = 57, P = -552 (--> (2, 23) etc)
> With the recurrence relation :
> Mk+1 = 1151*Mk + 120*Pk + 635
> Nk+1 = 11040*Mk + 1151*Pk + 6095
>
> That is restricting to >0 solution and merging :
> (2, 23) (57, 551) (515, 4944) (5697, 54648) (132362, 1269576)
> (1186680, 11382239) (13115642, 125800823) (304698417, 2922564551)
> ...
>
> You can do this for arbitrary chosen N, as explained in the link given
> by Tim Little.
> My example gives a case for which the "1-d" in the generalized Pell's
> equation in the link results into *several* fundamental solutions.
>
> Regards.
>
> --
> Philippe C., mail : chephip, with domain  free.fr
> site :http://mathafou.free.fr/  (mathematical recreations)


Thank you Tim,and Philippe



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