There are no solutions whenever k=4,5mod9k=4,5mod9 .
This holds since it is not difficult to verify that for every integer nn , we have n3=0,±1mod9n3=0,±1mod9 .
Up till this year, there were just two values for kk in the range 1..1001..100 for which it was not known if there were a solution. These values are k=33,42k=33,42 .
