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Euler's Sum of Powers Conjecture
Euler conjectured that at least
thpowers are required for
to provide a sum that is itself an
thpower. 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 Landeret al.(1967). Three 10.6.6 solutions were found by N. Kuosa (with
), displacing the 10.7.7 (
solution of Moessner (1939).
See also
Diophantine Equation--4th Powers,Diophantine Equation--6th Powers,Diophantine Equation--7th Powers,Diophantine Equation--8th Powers,Diophantine Equation--9th Powers,Diophantine Equation--10th Powers,Diophantine Equation--nth Powers,Euler Quartic ConjectureExplore with Wolfram|Alpha
References
Dutch, S. "Power Page: Euler's Conjecture."http://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 Mathematica48, 68-69, 1942.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers."http://euler.free.fr.Moessner, A. "Einige Numerische Identitaten."Proc. Indian Acad. Sci. Sect. A10, 296-306, 1939.Subba Rao, K. "On Sums of Sixth Powers."J. London Math. Soc.9, 172-173, 1934.Referenced on Wolfram|Alpha
Euler's Sum of Powers ConjectureCite this as:
Weisstein, Eric W. "Euler's Sum of Powers Conjecture."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/EulersSumofPowersConjecture.html