From bbrock@pepperdine.eduThu Aug 29 15:53:00 1996
Date: Thu, 29 Aug 1996 08:27:04 -0400
From: Bradley Brock <bbrock@pepperdine.edu>
To: NMBRTHRY@listserv.nodak.edu
Subject: Re: x^a + y^b = z^c

> Conjecture:
>
> X^a + Y^b = Z^c has no solutions for coprime naturals X,Y,Z when each
> exponent a,b,c > 2.

To answer this question one need only quote Granville's
post to this list from 2 1/2 years ago.
The bottom line is that there are no known solutions
and that for fixed a, b, & c, there can be at most
finitely many.

Brad
--
From: Andrew Granville <andrew@sophie.math.uga.edu>
Subject:      x^p+y^q=z^r
Comments: To: NmbrThry@vm1.nodak.edu
To: Multiple recipients of list NMBRTHRY <NMBRTHRY@VM1.NoDak.EDU>

In reply to a question of Dave Davis on x^p+y^q=z^r:

Henri Darmon and I have been looking at integer solutions to
the above equations. It is trivial to find lots for many
different p,q,r. For example, the following parametric
solution exists for exponents p,p,p+1:
  (ac)^p+(bc)^p = c^(p+1) where c = a^p+b^p.
To rid us of such upstart solutions, let us restrict
ourselves to solutions where x and y are coprime -- what we
will call `proper solutions'. It is `well-known'(see Dickson) that
1) The above equation has inf many proper solns whenever 1/p+1/q+1/r>1.
2) The above equation has no proper solns whenever 1/p+1/q+1/r=1 except
                  3^2-2^3=1^6

What Darmon and I have proved is:

If 1/p+1/q+1/r < 1 then there are only finitely many proper solutions
to x^p+y^q=z^r. The main tool we use is Faltings' theorem (nee Mordell's
conjecture); which we apply via an appropriate descent argument.
A preprint will soon be available (send email to me for this preprint).

Small examples of proper solutions to x^p+y^q=z^r with 1/p+1/q+1/r < 1
are:  1+2^3=3^2, \ \ 2^5+7^2=3^4, \ \ 7^3 + 13^2 = 2^9, \ \
      2^7+17^3=71^2, \ \ 3^5+11^4=122^2

Extraordinarily large solutions have been found recently by
Beukers and Zagier:
17^7 + 76271^3 = 21063928^2,   \ \ 1414^3 + 2213459^2 = 65^7, \ \
9262^3 + 15312283^2 = 113^7,   \ \ 3^8 + 96222^3 = 30042907^2, \ \
           33^8 + 1549034^2 = 15613^3.

It amazes me that there is such a gap between the two sets of solutions.
It may be that these are all of the solutions to the above, but I
thought that before Beukers and Zagier got involved, so maybe I am
wrong again. Can anyone compute some new examples ? That would be
very interesting.
    Andrew Granville            andrew@sophie.math.uga.edu