On Equal Sums of Like Powers (Euler's Conjecture)

In 1769, Euler, by generalising Fermat's Last Theorem, conjectured that ``it is impossible to exhibit three fourth powers whose sum is a fourth power'', ``four fifth powers whose sum is a fifth power, and similarly for higher powers''. The first counterexample to the conjecture, was found in 1966 after a systematic computer search:
27⁵+84⁵+110⁵+133⁵ =144⁵ More recently, Euler's conjecture was also disproved for fourth powers, and a method of generating an infinity of solutions to the equation A⁴+B⁴+C⁴=D⁴ is described by (Elkies, 1988) It is not known whether there are any counterexamples to Euler's conjecture for powers higher than the fifth. In this page, we consider a more general form of the above, that is, the diophantine equation
x₁^k + x₂^k + ... x_m^k = y₁^k + y₂^k + ... y_n^k, (1) and we are looking for positive integer solutions in x_i, y_j, for given k, m, n Obviously, we seek for non-trivial solutions (that is, solutions where there is no x_i equal to any y_j and vice versa).

We are especially concerned with the particular problem of finding the least n for which the above equation, for given k, m, is solvable.

In the following, adopting the notation from (Lander et al, 1967) we refer to (1) as (k.m.n).

Table showing the least n for which a solution to (1) is known.
Solutions for k=2.
Solutions for k=3.
Solutions for k=4.
Solutions for k=5.
Solutions for k=6.
Solutions for k=7.
Solutions for k=8.
Solutions for k=9.
Solutions for k=10.

References

A more up-to-date page with new results.
Noam D. Elkies. On A⁴+B⁴+C⁴=D⁴. Mathematics of Computation, 51, 1988, pp. 825-835.
Lander, Parkin, and Selfridge. A Survey of Equal Sums of Like Powers. Mathematics of Computation, 21, 1967, pp. 446-459.
Rizos Sakellariou. Solving Diophantine Equations on a Network of Workstations. In P.Sloot, M.Bubak, B.Hertzberger (Eds.), High-Performance Computing and Networking Europe, Lecture Notes in Computer Science 1401, Springer-Verlag, 1998, pp. 896-897.