Euler conjectured that at least th powers are required for to provide a sum that is itself an th power. The conjecture was disproved by Lander and Parkin (1967) with the counterexample
Ekl (1998) defined an extended Euler conjecture that there are no solutions to the Diophantine equation
with and not necessarily distinct, such that . Defining
over all known solutions to equations, this conjecture asserts that . There are no known counterexamples to this conjecture (Ekl 1998). The following table gives the smallest known values of for small .
|5||5.1.4||0||Lander et al. (1967)|
|6||6.3.3||0||Subba Rao (1934)|
|8||8.3.5||0||S. Chase (Meyrignac)|
|8||8.4.4||0||N. Kuosa (Nov. 9, 2006; Meyrignac)|
|9||9.5.5||1||Ekl 1997 (Meyrignac)|
|10||10.6.6||2||N. Kuosa (2002; Meyrignac)|
S. Chase found a 8.3.5 () solution that displaced the 8.5.5 () solution of Letac (1942). In 2006, N. Kuosa found an 8.4.4 solution with . Ekl (1996, 1998) found 9.4.6 and 9.5.5 solutions (both with ), displacing the 9.6.6 () solution of Lander et al. (1967). Three 10.6.6 solutions were found by N. Kuosa (with ), displacing the 10.7.7 ( solution of Moessner (1939).