Question

Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

Answer 1

I would assume that you want (x,y,z) to be an integer?
then:
k = 1 ----> (1,0,0) or (0,1,0) or (0,0,1)
k = 2 ----> (1,1,0) or .....
k = 3 ---> (1,1,1)
k = 4 ---- breakdown!
k = 73 ---> (4,1,2) or ....
if (x,y,z) is from the set of real numbers, you have an infinite number of cases
e.g.
k = 10 ---> (2,1,1), that's nice, or ( 3^(1/3), 1, 6^(1/3) ) or .....
k = 95 ----> (3,4, 4^(1/3)) or ....

Answer 2

Further to your question:
Here are the results of the values of k, x, y, z in groups of 4 for the values of k produced by integers x,y, and z from -50 to 50

1 15 1 -15 2 7 -5 -6 3 4 4 -5 6 2 -1 -1 7 2 0 -1 8 15 33 -34 9 2 1 0 10 4 -3 -3 11 3 -2 -2 12 10 7 -11 15 2 2 -1 16 14 -10 -12 17 2 2 1 18 3 -1 -2 19 3 0 -2 20 3 1 -2 21 -11 16 -14 24 8 8 -10 25 3 -1 -1 26 3 0 -1 27 15 3 -15 28 14 13 -17 29 13 18 -20 34 5 5 -6 35 14 -8 -13 36 4 -1 -3 37 4 0 -3 38 4 1 -3 43 12 8 -13 44 8 -5 -7 45 4 2 -3 46 3 3 -2 47 7 6 -8 48 4 -2 -2 53 5 -2 -4 54 12 -7 -11 55 10 -6 -9 56 4 0 -2 57 4 1 -2 60 5 -1 -4 61 5 0 -4 62 5 1 -4 63 7 -4 -6 64 15 4 -15 65 4 1 0 66 4 1 1 69 5 2 -4 70 11 20 -21 71 12 23 -24 72 9 7 -10 73 4 2 1 79 -19 35 -33 80 8 -6 -6 81 12 12 -15 82 14 -11 -11 83 6 -2 -5 88 6 -4 -4 89 6 6 -7 90 13 26 -27 91 6 0 -5 92 9 -5 -8 93 7 -5 -5
96 14 20 -22 97 5 -1 -3 98 14 9 -15 99 6 2 -5 100 7 -3 -6
sorry about the spacing

interpretation:
e.g. 73 4 2 1 ----> 73 = 4^3 + 2^3 + 1^3

if we restrict our x,y, and z to only whole numbers, we have only a few:
1 1 0 0 2 1 1 0 3 1 1 1 8 2 0 0 9 2 1 0 10 2 1 1 16 2 2 0 17 2 2 1 24 2 2 2 27 3 0 0 28 3 1 0 29 3 1 1 35 3 2 0 36 3 2 1 43 3 2 2 54 3 3 0 55 3 3 1 62 3 3 2 64 4 0 0 65 4 1 0 66 4 1 1 72 4 2 0 73 4 2 1 80 4 2 2 81 3 3 3 91 4 3 0 92 4 3 1 99 4 3 2

Answer 3

42?
I guessed