These pages are no longer maintained. They refer to results collected around 1994-1995 using idle time on a network of SUN workstations. The most interesting results (ie, those that were going beyond what was then known) were presented much later as a poster at HPCNE'98 (Published by Springer in Lecture Notes in Computer Science, vol. 1401, Apr. 1998), which is available here. A report (ref.5 in the paper) containing all the results found during this exercise is given here (sorry it is a bit messy!). For the current status, the definitive place to check is Jean-Charles Meyrignac's site.

On Equal Sums of Like Powers and Related Problems

These pages provide information on a class of problems related to the diophantine equation
x1k + x2k + ... xmk = y1k + y2k + ... ynk, (1)
where xi, yj integers, and k is greater than or equal to zero. The problem is that of finding non-trivial solutions (that is, solutions where there is no xi equal to any yj and vice versa) of the above equation.

Equations of this form have a long history. It can be seen that for m=1, n=2, Equation (1) has no solutions for k > 2 according to Fermat's Last Theorem. In 1769, this was generalised by Euler who conjectured that, for m=1, there are no solutions with k less than n. The first counterexample was found in 1966.

Rizos Sakellariou, 1998.