[seqfan] Re: Help needed

[Richard Mathar]

aj> From seqfan-bounces at list.seqfan.eu Fri Feb  5 14:02:48 2010
aj> Date: Fri, 05 Feb 2010 13:30:37 +0100
aj> From: Artur <grafix at csl.pl>
aj> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>, 
aj> 	Max Alekseyev <maxale at gmail.com>, zak seidov <zakseidov at yahoo.com>, 
aj> 	"Hasl >> Maximilian Hasler" <maximilian.hasler at gmail.com>
aj> Subject: [seqfan]  Help needed
aj> ..
aj> We are looking for such x that
aj> (37 + 2 x + x^2)/(-3 + 4 x) is perfect square
aj> ..
aj> How to proof that set x={2,7,32} is finite or find other x ?

The proposal is to solve the correponding quadratic equation for x,
baptizing the perfect square s^2:
x^2+2x+37 = s^2*(4x-3)
x^2+(2-4s^2)*x+37+3s^2 = 0
x = (1-2s^2)+-sqrt( 4s^4-7*s^2-36 )
then to look at the discriminant 4s^4-7s^2-36 which needs to be a square, say
4s^4-7*s^2-36  = k^2
and transform this into diophantine form by eliminating the linear term,
(2s^2)^2 -(7/2)*(2s^2)-36 = k^2
(2s^2)^2 -(7/2)*(2s^2)-36 = k^2
(2s^2-7/4)^2+49/16-36 = k^2
(8s^2-7)^2+49-576 = (4k)^2
(8s^2-7)^2-527 = (4k)^2
(8s^2-7)^2-(4k)^2 = 527
And this is hopefully covered as x^2-y^2 = q by the literature.

Richard Mathar