How do you find the sum of x³+y³+z³=k, for each k from 1 to 100.

Question

Have a hard time figuring it out, and I have a 97-100 in Algebra such a shame.

Any one I would love to get help!!

Answer 1

I will assume that k will be a whole number,
Not all values of k from 1 to 100 can be obtained with whole numbers of
x, y , and z
I will also assume that we are staying away from negative values of x, y, z

If y^3 and z^3 are zero, then the largest x possible for x^3 ≤ 110 is x = 4
(4^3 = 64 and 5^3 = 125, x = 5 is too big)
The same is true for y if both x and z are zero etc

x^3 + y^3 + z^3 is symmetric, that is, for any given value of x, y , and z
we could interchange them to get the same result
e.g. 2^3 + 3^3 + 1^3 = 3^3 + 1^3 + 2^3 =
So all we need is 3 cubes whose sum ≤ 100
they are:

possible triples: each time checking if the sum of cubes ≤ 100
1 0 0 --- 3 of them
1 0 1 --- 3 of them
1 0 2 --- 3 of them
1 0 3 --- 3 of them
1 0 4 --- 3 of them
1 1 0 --- 3 of them
1 1 1 --- 1 of them
1 1 2 --- 3 of them
1 1 3 --- 3 of them
1 1 4 --- 3 of them
1 2 0 --- 3 of them
1 2 2 --- 3 of them
1 2 3 --- 6 of them
1 2 4 --- 6 of them
1 3 0 --- 6 of them
1 3 3 --- 3 of them
1 3 4 --- 6 of them
2 2 2 --- 1 of them
2 2 3 --- 3 of them
2 2 4 --- 3 of them
2 3 3 --- 3 of them
3 3 3 --- 1 of them , I count a total of 72

I avoided duplicates,
e.g. skipped 2 2 1 near the end since already had it at 1 1 2

I hope I got all of them, tried to be systematic about it.
check if I missed any.

If the question was to actually get the sum of x^3 + y^3 + z^3,
you would have a lot of tedious busy-work ahead, don't think that is
what was asked.

Answer 2

In google paste:

cuemath algebra/sum-of-cubes-of-n-natural-numbers/

When you see list of results click on;

Sum of Cubes of n Natural Numbers - Formula, Proof, Examples

You will find very detail explanation.