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Most of the theoretical stuff on this part of my website is pertinent, but in terms of actual results, I am hopelessly out of date in most areas.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 8th, 22nd, 23rd and 24th powers (see Appendix A2), and will continue to work on these on and off. The major stumbling block for 8th 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.

DETAILED INTRODUCTION

Chapters 1 to 8 are on the following topics :

1. Euler's Conjecture
2. This is an accurate if out-of-date history of the subject.

3. Algorithms & Technical Background
4. 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.

5. Extended Euler Conjecture
6. When LHS can have more than one power.

7. Unequal Powers
8. Extending basic idea to expressing a power in terms of sums of another power.

9. Extending Unequal Powers
10. As with Chapter 3, considering multiple power sums on both sides.

Combined Results Only page for Chapters 4 & 5

11. Counting Solutions
12. An old, old topic of mine, counting solutions for small powers

13. Waring's Problem
14. Major article suggesting ideas on lower the bounds for G(n).

15. Extended Waring Problem

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

Last updated : 18/07/2003