Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.
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 ....
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
42?
I guessed
To solve this problem, we need to find values of x, y, and z such that the equation x³ + y³ + z³ = k holds true for each value of k from 1 to 100.
A brute-force approach would be to iterate through all possible combinations of x, y, and z, and check if the equation holds true for each value of k. However, this approach is computationally intensive and not feasible for large values of k.
A more efficient approach is to use a technique called "number theory" to solve the equation. We can take advantage of a property known as Euler's sum of powers conjecture, which states that any positive integer can be expressed as the sum of three cubes. Based on this conjecture, we can generate solutions for x, y, and z that satisfy the equation.
Here's an outline of the algorithm to find solutions for x, y, and z:
1. Initialize an empty list to store solutions.
2. Iterate through all possible combinations of x, y, and z within a given range, such as x = -100 to 100, y = -100 to 100, and z = -100 to 100. This range can be adjusted based on the required precision or the expected range of values for x, y, and z.
3. Calculate the value of k for each combination using the equation k = x³ + y³ + z³.
4. If the calculated k value matches the target value (from 1 to 100), add the combination (x, y, z) to the list of solutions.
5. Repeat steps 2 to 4 for all possible combinations.
6. The resulting list of solutions will contain multiple sets of (x, y, z) values that satisfy the equation x³ + y³ + z³ = k for each value of k from 1 to 100.
Note that this algorithm may generate multiple solutions for each value of k. The number of solutions may vary depending on the value of k. For example, some values of k may have unique solutions, while others may have multiple solutions.
Keep in mind that the Euler's sum of powers conjecture has not been proven to hold true for all integers, but it has been tested for a vast range of numbers and found to be true in almost all cases. Therefore, this algorithm provides a practical approach to finding solutions for x, y, and z in most cases.