Add Euler's totient function implementation - Implements φ(n) using prime factorization method

- Includes comprehensive tests for small numbers, primes, prime powers, and larger values
- All tests pass and follows project naming conventions

Author: triuyen

src/number_theory/euler_totient.rs

@@ -0,0 +1,70 @@
+pub fn euler_totient(n: u64) -> u64 {
+    if n == 1 {
+        return 1;
+    }
+
+    let mut result = n;
+    let mut num = n;
+    let mut p = 2;
+
+    // Find  all prime factors and apply formula
+    while p * p <= num {
+        // Check if p is a divisor of n
+        if num % p == 0 {
+            // If yes, then it is a prime factor
+            // Apply the formula: result = result * (1 - 1/p)
+            while num % p == 0 {
+                num /= p;
+            }
+            result -= result / p;
+        }
+        p += 1;
+    }
+
+    // If num > 1, then it is a prime factor
+    if num > 1 {
+        result -= result / num;
+    }
+
+    result
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn test_small_numbers() {
+        assert_eq!(euler_totient(1), 1);
+        assert_eq!(euler_totient(2), 1);
+        assert_eq!(euler_totient(3), 2);
+        assert_eq!(euler_totient(4), 2);
+        assert_eq!(euler_totient(5), 4);
+        assert_eq!(euler_totient(6), 2);
+    }
+
+    #[test]
+    fn test_prime_numbers() {
+        // For prime p, φ(p) = p - 1
+        assert_eq!(euler_totient(7), 6);
+        assert_eq!(euler_totient(11), 10);
+        assert_eq!(euler_totient(13), 12);
+        assert_eq!(euler_totient(17), 16);
+    }
+
+    #[test]
+    fn test_prime_powers() {
+        // For prime power p^k, φ(p^k) = p^(k-1) * (p-1)
+        assert_eq!(euler_totient(9), 6); // 3^2, φ(9) = 3^1 * 2 = 6
+        assert_eq!(euler_totient(25), 20); // 5^2, φ(25) = 5^1 * 4 = 20
+        assert_eq!(euler_totient(8), 4); // 2^3, φ(8) = 2^2 * 1 = 4
+    }
+
+    #[test]
+    fn test_larger_numbers() {
+        assert_eq!(euler_totient(10), 4);
+        assert_eq!(euler_totient(12), 4);
+        assert_eq!(euler_totient(100), 40);
+        assert_eq!(euler_totient(1000), 400);
+    }
+}

src/number_theory/mod.rs

@@ -1,5 +1,7 @@
 mod compute_totient;
+mod euler_totient;
 mod kth_factor;
 
 pub use self::compute_totient::compute_totient;
+pub use self::euler_totient::euler_totient;
 pub use self::kth_factor::kth_factor;