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

1,1

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

References

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

Links

Table of n, a(n) for n=1..43.

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)

Example

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]

Crossrefs

Cf. A063923, A063922, A003828, A175610.

Sequence in context: A008436 A250773 A262245 * A125841 A320457 A154051

Adjacent sequences: A134338 A134339 A134340 * A134342 A134343 A134344

Keyword

nonn

Author

Artur Jasinski, Oct 21 2007

Extensions

Incorrect formula removed by Jianing Song, Jan 24 2020

Status

approved