# Euler’s Factorization method

Given a number N, the task is to find the factors of N.

Examples:

Input: N = 1000009
Output: 293 3413
Explanation:
293 * 3413 = 1000009

Input: N = 100000
Output: 800 125
Explanation:
800 * 125 = 100000

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Euler’s Factorization method: Euler’s factorization method works on the principle that all the numbers N which can be written as the sum of two powers in two different ways can be factored into two numbers, (i.e) N = A2 + B2 = C2 + D2 where A != C and A != D, then there exists two factors for N.

Working of the algorithm: Let N be the number for which we need to find the factors.

1. So, initially, we need to find two ways to represent N as the sum of powers of two numbers.
```N = A2 + B2
N = C2 + D2
Therefore,
N = A2 + B2 = C2 + D2
```
2. Now, the algebraic operations are performed on the above equation to convert the equations as:
```N = A2 + B2 = C2 + D2
-> N = A2 - C2 = D2 - B2
-> N = (A - C)(A + C) = (D - B)(D + B)
```
3. Let K be the GCD of (A – C) and (D – B). So,
```A - C = K * L
D - B = K * M
where GCD(L, M) is 1.
```
4. Clearly, L = (A – C) / K and M = (D – B)/K. On substituting this in the initial equation:
```N = K * L * (A + C) = K * M * (D + B)
-> L * (A + C) = M * (D + B)
-&gtl (A + C)/(D + B) = M/L
```
5. Therefore:
```(A + C) = M * H
(D + B) = L * H
where,
H = gcd((A + C), (D + B))
```
6. Let Q = (K2 + H2)(L2 + M2).
```-> ((KL)2 + (KM)2 + (HL)2 + (HM)2)
-> ((A - C)2 + (D - B)2 + (D + B)2 + (A + C)2)
-> ((2 * A)2 + (2 * B)2 + (2 * C)2 + (2 * D)2)
-> 4 * N
```
7. Therefore,
```N = ((K/2)2 + (H/2)2)(L2 + M2)
```

Such that there exists a pair K and H which are both even numbers.

Let’s visualize the above approach by taking an example. Let N = 221.

1. 221 = 112 + 102 = 52 + 142
2. From the above equation:
```A = 11 - 5 = 6
B = 11 + 5 = 15
C = 14 - 10 = 4
D = 14 + 10 = 24
```
3. Therefore, the above values can be used to compute the values of K, H, L and M.
```K = GCD(6, 4) = 2
H = GCD(16, 24) = 8
L = GCD(6, 24) = 3
M = GCD(16, 4) = 2
```
4. Therefore:
```221 = ((2/2)2 + (8/2)2) * (32 + 22)
221 = 17 * 13
```

Approach: In order to implement the above approach, the following steps are computed:

1. Find the sum of squares by iterating a loop from 1 to sqrt(N) because no factor exists between [sqrt(N), N] apart from N and find two pairs whose sum of squares is equal to N.
2. Store the values in A, B, C, D.
3. Find the values of K, H, L and M using the formula mentioned in the above approach.
4. Use the values of K, H, L and M to find the factors. Check the pair where both the numbers is even and divide them in half and find the factors.

Below is the implementation of the above approach:

 `// C++ program to implement Eulers ` `// Factorization algorithm ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to return N as the sum of ` `// two squares in two possible ways ` `void` `sumOfSquares(``int` `n, vector >& vp) ` `{ ` `    ``// Iterate a loop from 1 to sqrt(n) ` `    ``for` `(``int` `i = 1; i <= ``sqrt``(n); i++) { ` ` `  `        ``// If i*i is square check if there ` `        ``// exists another integer such that ` `        ``// h is a perfect square and i*i + h = n ` `        ``int` `h = n - i * i, h1 = ``sqrt``(h); ` ` `  `        ``// If h is perfect square ` `        ``if` `(h1 * h1 == h) { ` ` `  `            ``// Store in the sorted way ` `            ``int` `a = max(h1, i), b = min(h1, i); ` ` `  `            ``// If there is already a pair ` `            ``// check if pairs are equal or not ` `            ``if` `(vp.size() == 1 && a != vp.first) ` `                ``vp.push_back(make_pair(a, b)); ` ` `  `            ``// Insert the first pair ` `            ``if` `(vp.size() == 0) ` `                ``vp.push_back(make_pair(a, b)); ` ` `  `            ``// If two pairs are found ` `            ``if` `(vp.size() == 2) ` `                ``return``; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to find the factors ` `void` `findFactors(``int` `n) ` `{ ` ` `  `    ``// Get pairs where a^2 + b^2 = n ` `    ``vector > vp; ` `    ``sumOfSquares(n, vp); ` ` `  `    ``// Number cannot be represented ` `    ``// as sum of squares in two ways ` `    ``if` `(vp.size() != 2) ` `        ``cout << ``"Factors Not Possible"``; ` ` `  `    ``// Assign a, b, c, d ` `    ``int` `a, b, c, d; ` ` `  `    ``a = vp.first; ` `    ``b = vp.second; ` ` `  `    ``c = vp.first; ` `    ``d = vp.second; ` ` `  `    ``// Swap if a < c because ` `    ``// if a - c < 0, ` `    ``// GCD cant be computed. ` `    ``if` `(a < c) { ` `        ``int` `t = a; ` `        ``a = c; ` `        ``c = t; ` `        ``t = b; ` `        ``b = d; ` `        ``d = t; ` `    ``} ` ` `  `    ``// Compute the values of k, h, l, m ` `    ``// using the formula mentioned ` `    ``// in the approach ` `    ``int` `k, h, l, m; ` `    ``k = __gcd(a - c, d - b); ` `    ``h = __gcd(a + c, d + b); ` `    ``l = (a - c) / k; ` `    ``m = (d - b) / k; ` ` `  `    ``// Print the values of a, b, c, d ` `    ``// and k, l, m, h ` `    ``cout << ``"a = "` `<< a ` `         ``<< ``"\t\t(A) a - c = "` `<< (a - c) ` `         ``<< ``"\t\tk = gcd[A, C] = "` `         ``<< k << endl; ` ` `  `    ``cout << ``"b = "` `<< b ` `         ``<< ``"\t\t(B) a + c = "` `<< (a + c) ` `         ``<< ``"\t\th = gcd[B, D] = "` `         ``<< h << endl; ` ` `  `    ``cout << ``"c = "` `<< c ` `         ``<< ``"\t\t(C) d - b = "` `<< (d - b) ` `         ``<< ``"\t\tl = A/k = "` `         ``<< l << endl; ` ` `  `    ``cout << ``"d = "` `<< d ` `         ``<< ``"\t\t(D) d + b = "` `<< (d + b) ` `         ``<< ``"\t\tm = c/k = "` `         ``<< m << endl; ` ` `  `    ``// Printing the factors ` `    ``if` `(k % 2 == 0 && h % 2 == 0) { ` `        ``k = k / 2; ` `        ``h = h / 2; ` ` `  `        ``cout << ``"Factors are: "` `             ``<< ((k) * (k) + (h) * (h)) ` `             ``<< ``" "` `<< (l * l + m * m) ` `             ``<< endl; ` `    ``} ` `    ``else` `{ ` `        ``l = l / 2; ` `        ``m = m / 2; ` ` `  `        ``cout << ``"Factors are: "` `             ``<< ((l) * (l) + (m) * (m)) ` `             ``<< ``" "` `<< (k * k + h * h) ` `             ``<< endl; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 100000; ` ` `  `    ``findFactors(n); ` ` `  `    ``return` `0; ` `} `

Output:

```a = 316        (A) a - c = 16        k = gcd[A, C] = 8
b = 12        (B) a + c = 616        h = gcd[B, D] = 56
c = 300        (C) d - b = 88        l = A/k = 2
d = 100        (D) d + b = 112        m = c/k = 11
Factors are: 800 125
```

Complexity Analysis:
Time Complexity: O(sqrt(N)), where N is the given number
Space Complexity: O(1)

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