# Sum of Two Cubes Example: Finding Cube Pairs

## Assignment:

You have a number x – find a couple of pairs that can represent x as a sum of two cubes.

Example:

$x = a^3 + b^3 = c^3 + d^3$

$32832 = (4^3 + 32^3)$ or $(18^3 + 30^3)$
$39312 = (2^3 + 34^3)$ or $(15^3 + 33^3)$
$40033 = (9^3 + 34^3)$ or $(16^3 + 33^3)$

This would be the solutions to the Diophantine equation, $a^3 + b^3 = c^3 + d^3$

Any number x that satisfies the demand will have two pairs $(a, b)$ and $(c, d)$ such that $a, b, c$ and $d$ are all less than $x1/3$. We should create an additional array of size $x1/3$. Every index $i$ in the array will contain the value that equals to the cube of $array[i] = i3$. Now the problem reduces to the search of the pair of elements in the sorted array whose sum is equal to given $x$.

C++ implementation:

```#include <iostream>
#include <cmath>
using namespace std;

void searchPairs(int n)
{
int root = pow(n, 1.0 / 3.0);

int *cube = new int[root + 1];

for (int i = 1; i <= root; i++)
cube[i] = i*i*i;

int a = 1;
int b = root;

while (a < b)
{
if (cube[a] + cube[b] < n) a++; else if (cube[a] + cube[b] > n)
b--;
else {
cout << "(" << a << ", " << b
<< ")" << endl;
a++; b--;
}
}
}

int main()
{
int n = 0;
cout << "Input the number you want to check: \n"; cin >> n;

searchPairs(n);

system("pause");
return 0;
}
```

Screenshots of the working program:

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