From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Extension to Fermat's Last?
Date: 24 Oct 1998 02:51:32 GMT

In article <70ofda$168$1@nnrp1.dejanews.com>,  <kchoi@iastate.edu> wrote:

>My question is: Do these numbers exist for every n > 2 such that
>a[1]^n + a[2]^n + ... + a[i]^n = z^n ?

[an earlier line, quoted from a previous post, specfied  i < n]

Two nonzero squares can sum to another.
Three nonzero cubes can sum to another, but not two.
Three nonzero fourth powers can sum to another, but not two. 
Four nonzero fifth powers can sum to another, but not two; It has not
	been decided whether or not three fifth powers can sum to another.
	I'll bet a nickel that they can't; who wants to take the bet?
What is the minimum number of positive sixth powers whose sum is another
	sixth power? It's more than two, certainly, but nothing else is
	known. I would not be surprised to hear that five sixth powers
	can sum to another, but in fact no one has ever even found SIX
	sixth powers whose sum is another sixth power (and some researchers
	are keenly interested in this.) What should be a comparably
	difficult problem, that of finding two equal sums of three sixth 
	powers, admits the easy solution 3,19,22; 10,15,23 found by Rao
	in 1934.

For even powers (or for odd powers with a positivity requirement added) 
one obtains a variety of different problems asking for solutions to
Sum x_i^n = Sum y_i^n, possibly with different numbers of summands on
the two sides. One can ask for parametric solutions (e.g. it is not
yet known for sure that there is not a parametric solution to
x^4+y^4+z^4=w^4 -- there are infinitely many distinct integer solutions).
One can ask for numbers multiply expressible as a sum of k n-th powers, or a
count of the number of representations as a sum of n-th powers, or a
count of the numbers in a given range which can be expressed as a sum
of n-th powers, or a count of the number of [positive] n-th powers needed
to express every [sufficiently large] integer as a sum of this many
n-th powers. One can ask for sets of integers the sum of whose n-th
powers are equal for more than one  n. And so on.

This is a part of Additive Number Theory, alive and well. 
	index/11PXX.html
Obviously the situation is better understood for lower powers. I can tell
you anything you want about sums of 1st powers :-)

dave