Added Legendre's Algorithm

docs/algorithms/Legendre's Algorithm/legendre-algo.md

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+---
+id: legendre-algo
+title: Finding Power of Factorial Divisor
+sidebar_label: Legendre's Formula
+---
+
+# Deep Dive into Finding the Power of a Factorial Divisor
+
+
+This algorithm is used to find the largest power x such that k^x divides n! , where n! is the factorial of a given number n , and k can be either a prime or composite number.
+
+
+## Why This Algorithm is Useful
+
+
+In number theory and combinatorics, determining the largest power of a divisor for factorials is crucial in problems involving:
+
+
+- **Divisibility in large products** (e.g., does n! divide a large power of some integer?)
+- **Prime factorization in combinatorial coefficients** (e.g., the power of a prime factor in binomial coefficients)
+- **Factorial simplifications in modular arithmetic** (useful in combinatorics, cryptography, etc.)
+
+
+This algorithm is especially valuable because calculating n! directly for large n is computationally infeasible, but by using Legendre's formula, we can determine the power of a divisor without computing the factorial itself.
+
+
+---
+
+
+## Understanding the Problem for Prime k 
+
+
+Let's first consider the case where k is a **prime number**.
+
+
+### Example Problem: Power of Prime Factor Dividing n! 
+
+
+Suppose we want to find the largest power of a prime k = 2 that divides n! , where n = 10:
+
+
+$$ 10! = 1 * 2 * 3 * . . . * 10 = 3628800 $$
+
+
+
+We need to find the highest power x such that 2^x divides 3628800.
+
+
+### Derivation of Legendre's Formula
+
+
+1. **Counting Divisors of Prime k :** 
+
+
+   The factorial $$ n! = 1 * 2 * 3 * n $$ contains various multiples of k, but not every number from 1 to n is divisible by k.
+
+
+2. **Step-by-Step Divisor Counting for Powers of k :**
+
+
+   - **First Power k :** Every k-th number (i.e., every multiple of k) in the product from 1 to n contributes a factor of k . There are $$\left\lfloor \frac{n}{k} \right\rfloor$$ such numbers.
+   - **Second Power k^2:** Every k^2 -th number contributes an additional factor of k, so there are $$\left\lfloor \frac{n}{k^2} \right\rfloor $$ such numbers.
+   - **Continuing for Higher Powers:** This pattern continues until $$ k^i > n $$.
+
+
+3. **Summing Up All Factors of k:**
+
+
+   By adding up the counts for each power of k , we get the largest power x such that k^x divides n! :
+
+
+$$
+   x = \left\lfloor \frac{n}{k} \right\rfloor + \left\lfloor \frac{n}{k^2} \right\rfloor + \left\lfloor \frac{n}{k^3} \right\rfloor + \cdots
+$$
+
+
+4. **Example Calculation \( k = 2 \):**
+
+
+   Let's compute this for n = 10 and k = 2 .
+
+
+   $$
+   x = \left\lfloor \frac{10}{2} \right\rfloor + \left\lfloor \frac{10}{4} \right\rfloor + \left\lfloor \frac{10}{8} \right\rfloor
+   $$
+
+
+   - $$\left\lfloor \frac{10}{2} \right\rfloor = 5 $$
+   - $$\left\lfloor \frac{10}{4} \right\rfloor = 2 $$
+   - $$\left\lfloor \frac{10}{8} \right\rfloor = 1 $$
+
+
+   Thus, $$ x = 5 + 2 + 1 = 8 $$. So, $$ 2^8 $$ is the largest power of 2 that divides 10!.
+
+
+---
+
+
+### Implementation for Prime k 
+
+
+The code to compute this for prime k is as follows:
+
+
+```cpp
+int fact_pow(int n, int k) {
+    int res = 0;
+    while (n) {
+        n /= k;
+        res += n;
+    }
+    return res;
+}
+```
+
+
+### Complexity Analysis
+
+
+This algorithm is $$O(\log_k n) $$, which is efficient, as it only needs to consider powers of k up to $$\log_k n$$.
+
+
+---
+
+
+### Extending to Composite $$k$$
+
+
+When k is composite, we need to factorize k and determine how many times each prime factor of k appears in n!.
+
+
+#### Example Problem: Power of Composite Divisor Dividing $$n! $$
+
+
+Suppose n = 10 and k = 12. We want to find the largest power $$x$$ such that $$12^x$$ divides 10!.
+
+
+1. **Factorize $$k = 12 $$:**
+
+
+   $$
+   12 = 2^2 \cdot 3^1
+   $$
+
+
+   We can see that k has prime factors 2 and 3 with respective powers 2 and 1.
+
+
+2. **Apply Legendre's Formula to Each Prime Factor:**
+
+
+   - For $$ k_1 = 2 $$, we previously found that the power of $$2 in 10! $$ is $$8$$.
+   - For $$k_2 = 3$$, using the formula, we get:
+
+$$
+
+     \left\lfloor \frac{10}{3} \right\rfloor + \left\lfloor \frac{10}{9} \right\rfloor = 3 + 1 = 4
+
+$$
+
+   Thus, the power of 3 in 10! is 4.
+
+
+3. **Calculate the Minimum Power Dividing $$k$$:**
+
+
+   - For $$ 2^2 $$, we need $$ \frac{8}{2} = 4 $$.
+   - For $$ 3^1 $$, we need $$\frac{4}{1} = 4$$ .
+
+
+   The minimum of these two values is 4 , so 12^4 is the largest power of 12 that divides 10!.
+
+
+#### Code Implementation for Composite $$k$$
+
+
+To implement this approach, we first factorize k and then use Legendre's formula for each factor.
+
+
+```cpp
+#include <iostream>
+#include <vector>
+#include <cmath>
+#include <climits>
+
+
+// Function to find the largest power of a prime p that divides n!
+int fact_pow(int n, int p) {
+    int res = 0;
+    while (n) {
+        n /= p;
+        res += n;
+    }
+    return res;
+}
+
+
+// Function to factorize k into prime factors and their powers
+std::vector<std::pair<int, int>> prime_factors(int k) {
+    std::vector<std::pair<int, int>> factors;
+    for (int i = 2; i * i <= k; i++) {
+        int count = 0;
+        while (k % i == 0) {
+            k /= i;
+            count++;
+        }
+        if (count > 0) {
+            factors.push_back({i, count});
+        }
+    }
+    if (k > 1) {
+        factors.push_back({k, 1});
+    }
+    return factors;
+}
+
+
+// Function to find the largest power of a composite k that divides n!
+int composite_fact_pow(int n, int k) {
+    std::vector<std::pair<int, int>> factors = prime_factors(k);
+    int result = INT_MAX;
+    for (auto &factor : factors) {
+        int prime = factor.first;
+        int power = factor.second;
+        int power_in_factorial = fact_pow(n, prime);
+        result = std::min(result, power_in_factorial / power);
+    }
+    return result;
+}
+
+
+int main() {
+    int n = 10; // Example value
+    int k = 12;  // Example composite number
+    std::cout << "Largest power of " << k << " that divides " << n << "! is: "
+              << composite_fact_pow(n, k) << std::endl;
+    return 0;
+}
+```
+
+
+### Use Cases and Applications
+
+
+This algorithm is widely useful in fields involving large combinatorial structures, especially when direct computation of n! is impractical:
+
+
+1. **Divisibility Testing:** In combinatorial problems where you need to check if n! divides a large power, this method efficiently finds the highest power.
+2. **Factorization in Modular Arithmetic:** In applications involving modular inverses or multiplicative groups.
+3. **Cryptography:** Particularly in algorithms like RSA or in solving Diophantine equations where factorial divisibility is often involved.
+
+
+---
+
+
+### Conclusion
+
+
+Legendre's algorithm is a highly efficient method for calculating the largest power of a divisor that divides a factorial. It plays a crucial role in number theory, combinatorics, and cryptography, where it provides an alternative to direct factorial computation, enabling efficient solutions for large numbers.
+
+
+