Euler's Sum of Powers Conjecture

Euler conjectured that at least n nth powers are required for n>2 to provide a sum that is itself an nth 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 k.m.n Diophantine equation


with a_i and b_i not necessarily distinct, such that m+n<k. Defining


over all known solutions to k.m.n equations, this conjecture asserts that Delta_k>=0. There are no known counterexamples to this conjecture (Ekl 1998). The following table gives the smallest known values of Delta_k for small k.

kmin. Delta_k soln.Delta_kreference
44.1.30Elkies (1988)
55.1.40Lander et al. (1967)
66.3.30Subba Rao (1934)
77.4.41Ekl (1996)
88.3.50S. Chase (Meyrignac)
88.4.40N. Kuosa (Nov. 9, 2006; Meyrignac)
99.5.51Ekl 1997 (Meyrignac)
1010.6.62N. Kuosa (2002; Meyrignac)

S. Chase found a 8.3.5 (Delta_8=0) solution that displaced the 8.5.5 (Delta_8=2) solution of Letac (1942). In 2006, N. Kuosa found an 8.4.4 solution with Delta_8=0. Ekl (1996, 1998) found 9.4.6 and 9.5.5 solutions (both with Delta_9=1), displacing the 9.6.6 (Delta_9=3) solution of Lander et al. (1967). Three 10.6.6 solutions were found by N. Kuosa (with Delta_(10)=2), displacing the 10.7.7 (Delta_(10)=4) 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 Conjecture

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Dutch, S. "Power Page: Euler's Conjecture.", 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 A^4+B^4+C^4=D^4." 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.", 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.

Referenced on Wolfram|Alpha

Euler's Sum of Powers Conjecture

Cite this as:

Weisstein, Eric W. "Euler's Sum of Powers Conjecture." From MathWorld--A Wolfram Web Resource.

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