Number of $x^3$ required to equal $2002^{2002}$. While working on math problems, I came upon a high-power summing problem, and got bogged down. The problem is as follows: What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \dots, x_{t}$ with $x_{1}^3$ + $x_{2}^3$ +...+ $x_{t}^3=2002^{2002}$. I think that it means to find the minimum value of t where t is the largest base among the sum of the cubes of these bases. However, I am completely lost, and don't know how to solve this type of problem. Can someone help? :) Also, this problem is from the mods section of the book, so maybe use mod 2002 or something?

$2002^{2002} =(2002^{667})^3. (10^3+10^3+1^3+1^3) $

$=(2002^{667}×10)^3+(2002^{667}×10)^3+(2002^{667}×1)^3+(2002^{667}×1)^3$

It seems $t=4$ is minimum as 2002 can be expressed as a minimum of 4 cubes.