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 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