From: tpiezas@uap.edu.ph (Tito Piezas III)
Subject: The Sum of n nth Powers Conjecture
Date: 8 Dec 2000 10:18:29 -0500
Newsgroups: sci.math
Summary: [missing]

Dec. 8, 2000

Euler had a related conjecture.  In 1772, he stated:

    "If a sum of r positive nth powers equal one nth power, then r is
greater than or equal to n."

The first counter-example was by Lander, Parkin, and Selfridge in 1966
when they showed an example of r=4, but n=5.

    27^5 + 84^5 + 110^5 + 135^5 = 144^5

So even Euler could be wrong.

Now, my interest is when we let r=n and phrase Euler's conjecture
differently.  The "sum of n nth powers conjecture" will be:

    "There exists a positive integer Z such that beyond Z there is no
one nth power equal to the sum of n nth powers."

For n=2,3,4,5

    2^2 + 3^2 = 5^2
    3^3 + 4^3 + 5^3 = 6^3
    30^4 + 120^4 + 272^4 + 315^4 = 353^4
    19^5 + 43^5 + 46^5 + 47^5 + 67^5 = 72^5

In 1967, Lander (et all) stated than no solutions were known for n>5. 
As late as 1994, R.K. Guy in "Sums of Like Powers: Euler's Conjecture"
stated the same thing, none found for n=6,7,8,9, etc.

But in Mar 20, 1999, Mark Dodrill found:

    127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 = 568^7

In Nov. 14, 2000, Maurice Blondot found:

     91^7 + 148^7 + 158^7 + 255^7 + 258^7 + 309^7 + 625^7 = 626^7

However, there is still no solution for n=6.

By the wording of our conjecture, at least up to this day, we've
pushed Z to Z=7.  The question to be asked:  How far can we push Z? 
Can we do it indefinitely, or we'll we reach a barrier at some point?

So, for those of you with programming skills, here's a chance to use
those idle computer time.  To find n=6, and any Z > 7, if any.  And
how do we prove or disprove the conjecture?

Note: Some of the material for this post is from the remarkable
website of Jean-Charles Meyrignac: euler.free.fr/index.htm (if you
discover anything, please post here as well as there) and Eric
Weinstein's former mathworld.wolfram.com, the termination of which I
am whole-heartedly against.


Sincerely,

Tito Piezas III
University of Asia & the Pacific