From: steiner@math.bgsu.edu (ray steiner)
Subject: Re: Help with Diophantine Eq's
Date: Sat, 18 Sep 1999 13:21:19 -0500
Newsgroups: sci.math
Keywords: exercises from Rosen

In article <17599f0b.86772295@usw-ex0108-058.remarq.com>, Bart Goddard 
<goddardbNOpuSPAM@concordia.edu> wrote:

> Hi,
> 
> Just to be clear, the background is this:
> 1. I'm writing the solution manual to 
>   Ken Rosen's new edition of his number
>   theory text.
> 
> 2.  I'm getting paid for it.
> 
> There are over 1200 exercises in this text
> and I have solved all of them.  (And corrected
> many of the dense set of errors in my 
> previous solutions manual.)  However:
> 
> There are three exercises left which I'm
> just too cross-eyed to finish.  In the 
> 3rd edition they are 11.1.7, 11.2.7 and
> 11.4.13. So I'm here, hat in hand, asking for
> a bit of charity. The last two are easy to state:
> 
> 11.2.7  Show that the Diophantine equation
> x^4-8y^4=z^2 has no solutions in the nonzero
> integers.
> 
> (I think the way to proceed here is to 
> paramaterize the solutions to x^2+2y^2=z^2
> and cook up a descent argument.)
> 
> 11.4.13 Show that the Diophantine equation
> x^4-2y^2=-1 has no nontrivial solutions.
> 
> (Here, I'm trying to show that the Pell's
> solution for x^2 can't be a square, but no
> luck so far.)
> 
> 11.1.7:  Let x_1 = 3, y_1 = 4, z_1 = 5 and
> 
> x_{n+1} = 3x_n + 2z_n +1
> y_{n+1} = 3x_n + 2z_n +2
> z_{n+1} = 4x_n + 3z_n +2
> 
> Show that any pythagorean triple  with y=x+1
> is one of these.
> 
> If anyone is bored and would like to help me
> out, I'd appreciate it. I don't have good access
> to this group, so please e-mail your kind
> thoughts to me at
> 
> goddardb@concordia.edu
> 
> Thanks in advance,
> Bart Goddard
The first two of these are done(using Pythagorean triples and infinite 
descent) in
the book CATALAN's CONJECTURE by Ribenboim. More generally, the only solution
of x^2-8y^4=z^4 is (x,y,z)= (3,1,1).
Hope this helps.
Regards,
Ray Steiner

-- 
steiner@math.bgsu.edu