Equations involving sums of powers

Considering positive integer solutions to equations involving sum of like powers

A number of loosely related investigations into various types of equation involving sums of powers, the number of elements in the sum being of indeterminate length.

SPECIAL NOTE (12/01/2003): Most of the theoretical stuff on this part of my website is pertinent, but in terms of actual results, I am hopelessly out of date in most areas. For an up-to-date state of affairs on all results pertaining to same power sums, always refer to the highly recommended website of Jean-Charles Meyrignac.

WHAT'S NEW :

18/07/2003 - at last some action! I have updated the tables of Euler lengths for 8^{th}, 22^{nd}, 23^{rd} and 24^{th} powers (see Appendix A2), and will continue to work on these on and off. The major stumbling block for 8^{th} powers was at x = 47, which took me several years to get round so solving.

29/01/2001 - updated version of Chapter 1, incorporating the use of lattice reduction algorithms by Crump, Childers et al to make huge inroads into the results for higher powers.

Chapters 1 to 8 are on the following topics :

- Euler's Conjecture
- Algorithms & Technical Background
- Extended Euler Conjecture
- Unequal Powers
- Extending Unequal Powers
- Counting Solutions
- Waring's Problem
- Extended Waring Problem

This is an accurate if out-of-date history of the subject.

This contains some of the algorithms used by me. For more generic descriptions in terms of implementing for a computer, look in the companion article McLean's Algorithms.

When LHS can have more than one power.

Extending basic idea to expressing a power in terms of sums of another power.

As with Chapter 3, considering multiple power sums on both sides.

**Combined Results Only page for Chapters 4 & 5**

An old, old topic of mine, counting solutions for small powers

Major article suggesting ideas on lower the bounds for G(n).

Tentative attempt to extend Waring's problem in the same way as Euler's conjecture.

URL : www.glasgowg43.freeserve.co.uk/sumintro.htm

Last updated : 18/07/2003