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.