From: steiner@math.bgsu.edu (ray steiner)
Subject: Re: x^3 + y^3 = 21 * z^3, integer solutions ?
Date: Fri, 03 Sep 1999 12:52:42 -0500
Newsgroups: sci.math,sci.math.num-analysis
Keywords: x^3+y^3=Az^3

In article <37CF89EE.AD9B6D2@ipb.uni-bonn.de>, Boudewijn Moonen 
<Boudewijn.Moonen@ipb.uni-bonn.de> wrote:

> Siegbert Steinlechner wrote:
> > 
> > Hello,
> > 
> > we know from Fermat's theorem, that
> > 
> >         x^3 + y^3 = z^3
> > 
> > doesn't have integer solutions.
> > 
> > But I've heard that e.g.
> > 
> >         x^3 + y^3 = 21 * z^3
> > 
> > has integer solutions.
> > 
> > Who knows about such solutions? How to get solutions?
> > The numbers x, y, z seem to be rather big; a simple search routine
> > in MATLAB didn't get results because of computation time and
> > finite presision.
> > 
> > S. Steinlechner
> 
> 
> For own computations, try PARI. It is available at
> 
>         http://hasse.mathematik.tu-muenchen.de/ntsw/pari/Welcome_e.html
> 
> -- 
>     Boudewijn Moonen
>     Institut fuer Photogrammetrie der Universitaet Bonn
>     Nussallee 15
>     
>     D-53115 Bonn
>     
>     GERMANY

Save the computations. If you look on the last page of THE THEORY OF 
IRRATIONALITIES
OF THE THIRD DEGREE by Delone and Faddeev(Eng. tr. AMS(1964)), you will find
a table of all basic solutions of x^3+y^3=Az^3 for A<=50. They state that
it is known that there are no nonzero solutions for A=21. 
There is also an article by Selmer in Acta Mathematica 85(1951), 203-362
on this subject. In particular, see Theorem VIII on p.301, due to Sylvester
and Pepin which says(among other things) that if A= q_1q_2 where q_1 and q_2
are primes and if the eqn is solvable, then q_1= 2(mod 9) and q_2=5(mod 9) 
This rules out 21.
Regards,
Ray Steiner

-- 
steiner@math.bgsu.edu