Euler's [sum of powers conjecture][1], if a sum of $k$th powers is a $k$th power, then the sum has at least $k$ terms.


  [1]: http://en.wikipedia.org/wiki/Euler%2527s_sum_of_powers_conjecture

Proposed by Euler in 1769.  Counterexample for $k=5$ found in 1966, for $k=4$ in 1986.