A sextic counterexample to Euler's sum of powers conjecture

Importance: Medium ✭✭
Author(s): Euler, Leonhard P.
Recomm. for undergrads: yes
Posted by: maxal
on: August 5th, 2007

\begin{problem} Find six positive integers $x_1, x_2, \dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist. \end{problem}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

\Def{Euler's sum of powers conjecture} states that for $k\geq 3$ the \Def{Diophantine equation} $\sum_{i=1}^{n} a_i^k = b^k$ does not have solutions in positive integers as soon as $n


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

\href[EulerNet: Computing Minimal Equal Sums Of Like Powers]{http://euler.free.fr/}

* indicates original appearance(s) of problem.