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
 |
(1)
|
Ekl (1998) defined an extended Euler conjecture that there are no solutions to the 
Diophantine equation
 |
(2)
|
with 
and 
not necessarily distinct, such that 
. Defining
 |
(3)
|
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 
.
 | min.  soln. |  | reference |
| 4 | 4.1.3 | 0 | Elkies (1988) |
| 5 | 5.1.4 | 0 | Lander et al. (1967) |
| 6 | 6.3.3 | 0 | Subba Rao (1934) |
| 7 | 7.4.4 | 1 | Ekl (1996) |
| 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).
." Math. Comput. 51, 828-838,
1988.Hoffman, P.