Find all three-digit numbers that are equal to the sum of the cubes of the digits.
As pointed out, a part of the problem you have raised refers to Armstrong numbers / narcissistic numbers, i.e., nn -digit numbers equal to the sum of nn -th power of its digits. Besides that, there is nothing really special about such numbers. For n=3n=3 , there are only 44 such numbers, 153,370,371153,370,371 and 407407 . There is no analytical means to solve those equations exactly, thus one has to rely on trial-and-error or brute force.
The other part, where a 4-digit number is equal to sum of third powers of its digits, does not have any solution. Consider, as you showed:
Adding, 10001000 on both sides, we have:
Clearly, for any d∈[1,9]d∈[1,9] , we have:
Conclusion: no solutions exist.
Hope this helps, whymenotsmart^m^, although guest's answer is alternable:)