From: Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>
Subject: Re: prime factorization of 2^22 + 1
Date: Fri, 26 Nov 1999 18:07:26 +0100
Newsgroups: sci.math
Keywords: Aurifeuillian factorizations of sums of powers

Jim Ferry wrote:

> Robin Chapman wrote:
> >
> > You might recall that x^2 - y^2 = (x - y)(x + y). It's also the
> > case (but far less well-known!) that 4x^4 + y^4 also has a nice
> > factorization.
>
> I didn't know that.  The *second* factorization, that is.  I did
> some experimenting and found that k^n x^(2n) + y^(2n) sometimes
> has non-trivial factorizations when k | n.  Examples below.  Is
> there a theory behind this, or at least something interesting to
> say about it?

 These are called Aurifeuillian factorizations. Look up
Brillhart, John; Lehmer, D. H.; Selfridge, J. L.; Tuckerman, Bryant;
Wagstaff, S. S., Jr. Factorizations of $b\sp n ±1$.
$b=2,3,5,6,7,10,11,12$ up to high powers.
American Mathematical Society,  1988.
    or
Riesel, Hans Prime numbers and computer methods for factorization.
Second edition.
Progress in Mathematics, 126. Birkhäuser Boston, Inc., Boston, MA, 1994

franz