A134341 - OEIS

%I #44 Aug 20 2023 10:49:30

%S 144,288,432,576,720,864,1008,1152,1296,1440,1584,1728,1872,2016,2160,

%T 2304,2448,2592,2736,2880,3024,3168,3312,3456,3600,3744,3888,4032,

%U 4176,4320,4464,4608,4752,4896,5040,5184,5328,5472,5616,5760,5904,6048,6192

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

%C 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

%C 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

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

%H L. J. Lander and T. R. Parkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1966-11654-3">Counterexample to Euler's conjecture on sums of like powers</a>, Bull. Amer. Math. Soc. 72 (6) (1966), p. 1079.

%H Burkard Polster, <a href="https://www.youtube.com/watch?v=AO-W5aEJ3Wg">Euler's and Fermat's last theorems, the Simpsons and CDC6600</a>, Mathologer video (2018).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture">Euler's sum of powers conjecture</a>

%H <a href="/index/Di#Diophantine">Index to sequences related to Diophantine equations</a> (5,1,4)

%e a(1) = 144 because 144^5 = 27^5 + 84^5 + 110^5 + 133^5;

%e 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]

%Y Cf. A063923, A063922, A003828, A175610.

%K nonn

%O 1,1

%A _Artur Jasinski_, Oct 21 2007

%E Incorrect formula removed by _Jianing Song_, Jan 24 2020