From: ribet@digel.berkeley.edu (Kenneth A. Ribet)
Newsgroups: sci.math.research
Subject: Re: generalized Fermat-type equations
Date: 17 Dec 1996 00:07:31 GMT

>J. M. Gandhi conjectured in 1964 that integer solutions (x,y,z) of

>	x^5 + y^5 = 2*z^5    

>must satisfy  x = y or x = -y, and likewise for
> 
>	x^7 + y^7 = 2*z^7.

>Have these conjectures been resolved by now?  (Surely they must!)
>References?  Please visit my web page

>	http://www.mathsoft.com/asolve/fermat/fermat.html

>for other problems involving the equation  x^n + y^n = c*z^n.

>					Steve Finch
>					sfinch@mathsoft.com

There's a 1952 paper of P. Denes on this subject, "Uber die
Diophantische Gleichung x^l+y^l=cz^l", which seems to resolve the
two equations that you list.  The problem becomes more difficult,
however, if you take a large prime number l in place of 5 or 7
(and keep c=2).

In his 1987 Duke Journal paper, Serre shows that the elliptic curve
methods which resolved the Fermat problem shed light on such
equations.  Serre gives a list of prime numbers which can be taken
as "c" -- for those, there are no solutions in non-zero x, y, and
z when l is a prime bigger than 2 and l is different from c.  Serre's
prime numbers "c" are all odd, and they lead to semistable elliptic
curves.  The choice c=2 leads to a non-semistable curve.  It's also
different in character because the equation x^l + y^l = 2 z^l does
have non-zero solutions, namely those with x=y=z!

Several years ago, I received a letter from an amateur mathematician
about such equations.  I wrote an article which will appear (some day)
in Acta Arithmetica.  It can be picked up (as a TeX file) via the
URL ftp://math.berkeley.edu/pub/Preprints/Ken_Ribet/euler.tex.  Basically,
I prove that x^l + y^l = 2 z^l has only the obvious non-zero solutions
when l is congruent to 1 mod 4.  My methods are pretty much the same
as those introduced previously by H. Darmon to deal with some other
Diophantine equations.

In a recent preprint, Darmon and L. Merel deal with the case where
l is congruent to 3 mod 4, thereby proving for all odd primes l
that the only non-zero solutions to x^l + y^l = 2 z^l are the
evident ones with x=y=z.  This theorem resolves a problem which
has been of interest at least since the 17th century!  You can
pick up the preprint (again in TeX format) as
http://www.math.mcgill.ca/~darmon/pub/Winding/paper.html.

-ken ribet