Euler's Sum of Powers Conjecture
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).
REFERENCES:
Dutch, S. "Power Page: Euler's Conjecture." https://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#eulercon.
Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755-1756, 1996.
Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.
Elkies, N. "On 
." Math. Comput. 51, 828-838, 1988.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 195, 1998.
Lander, L. J. and Parkin, T. R. "A Counterexample to Euler's Sum of Powers Conjecture." Math. Comput. 21, 101-103, 1967.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967.
Letac, A. Gazetta Mathematica 48, 68-69, 1942.
Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." https://euler.free.fr.
Moessner, A. "Einige Numerische Identitaten." Proc. Indian Acad. Sci. Sect. A 10, 296-306, 1939.
Subba Rao, K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172-173, 1934.
