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A sextic counterexample to Euler's sum of powers conjecture
Problem Find six positive integers
such that
or prove that such integers do not exist.
![$ x_1, x_2, \dots, x_6 $](/files/tex/7d2f648b8bcee43356c8f16ad89b1b99848de739.png)
![$$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$](/files/tex/ac6c0341dacf4c6f72d84950d5f4604df7778416.png)
Euler's sum of powers conjecture states that for the Diophantine equation
does not have solutions in positive integers as soon as
For
it corresponds to a particular case of Fermat Last Theorem and hence is true. For
and
, counterexamples to the Euler's sum of powers conjecture were found by N. Elkies in 1986 and L. J. Lander, T. R. Parkin in 1966 respectively. For
, no counterexamples are currently known.
* indicates original appearance(s) of problem.