A k-tetrahedral number (k-simplicial would be better terminology) is just the analogue of a triangular or tetrahedral number but in k dimensions. The nth k-tetrahedral number is T_k(n) = (n+k-1)(n+k-2)...(n+1)n / k! or T_k(n) = (n+k-1)_C_k .* Question: Can T_k(n) be an exact kth power for k > 2 (and n > 1) ??? The question arose after a cursory check of the first few tetrahedral numbers, none of which seem to be perfect cubes. Of course, triangular numbers can be perfect squares, like 36, 1225, etc. —Dan _____________ * The formula is easily proved by induction, since e.g. the nth tetrahedral number is just a stack of the first n triangular numbers. But is there a geometrical reason that it should be the number of size-k subsets of a set of n+k-1 objects?