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dushya
Posts:
31
Registered:
11/18/09


Re: An equation
Posted:
Aug 19, 2011 11:05 AM


On Aug 19, 1:48 pm, "Philippe 92" <nos...@free.invalid> wrote: > Tim Little wrote : > > > > > > > > > > > On 20110819, 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/ratiooftriangularnumbers > > 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 "1d" 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



