A134341 Numbers whose fifth powers have a partition as a sum of fifth powers of four positive integers.

144, 288, 432, 576, 720, 864, 1008, 1152, 1296, 1440, 1584, 1728, 1872, 2016, 2160, 2304, 2448, 2592, 2736, 2880, 3024, 3168, 3312, 3456, 3600, 3744, 3888, 4032, 4176, 4320, 4464, 4608, 4752, 4896, 5040, 5184, 5328, 5472, 5616, 5760, 5904, 6048, 6192

Offset: 1

Author

Artur Jasinski, Oct 21 2007

Keywords

• nonn

Comments

The only primitive terms (that is, in which the summands do not all have a common factor) known are 144 and 85359. - Jianing Song, Jan 24 2020

The paper by Lander and Parkin where they just give the first known counterexample to Euler's conjecture, 27^5 + 84^5 + 110^5 + 133^5 = 144^5, found using a CDC6600, is known as one of the shortest published proofs. - M. F. Hasler, Mar 11 2020

Examples

			a(1) = 144 because 144^5 = 27^5 + 84^5 + 110^5 + 133^5;
a(593) = 85359 because 85359^5 = 55^5 + 3183^5 + 28969^5 + 85282^5 = 4531548087264753520490799 (Jim Frye 2005). [Typo corrected by _Sébastien Palcoux_, Jul 05 2017]
		

References

• L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952, p. 648.

Links

• L. J. Lander and T. R. Parkin, Counterexample to Euler's conjecture on sums of like powers, Bull. Amer. Math. Soc. 72 (6) (1966), p. 1079.
• Burkard Polster, Euler's and Fermat's last theorems, the Simpsons and CDC6600, Mathologer video (2018).
• Wikipedia, Euler's sum of powers conjecture
• Index to sequences related to Diophantine equations (5,1,4)

Crossrefs

Cf. A063923, A063922, A003828, A175610.

Extensions

Incorrect formula removed by Jianing Song, Jan 24 2020