Merge pull request #251 from Priyangshuyogi/myone

Algorithms on Primality Test

Author: prateekiiest

Competitive Coding/Math/Primality Test/Fermat Method/README.md

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+Given a number n, check if it is prime or not.  This method is a probabilistic method and is based on below Fermat’s Little Theorem.
+
+Fermat's Little Theorem:
+If n is a prime number, then for every a, 1 <= a < n,
+
+a^n-1 ~ 1 mod (n)
+ OR 
+a^n-1 % n ~ 1 
+ 
+
+Example:
+
+         Since 5 is prime, 2^4 ≡ 1 (mod 5) [or 2^4%5 = 1],
+         
+         3^4 ≡ 1 (mod 5) and 4^4 ≡ 1 (mod 5). 
+
+         Since 7 is prime, 2^6 ≡ 1 (mod 7),
+         
+         3^6 ≡ 1 (mod 7), 4^6 ≡ 1 (mod 7) 
+         5^6 ≡ 1 (mod 7) and 6^6 ≡ 1 (mod 7).
+         
+We will take help of this Fermat's Little Theorem as a function to calculate a no is prime or not.         

Competitive Coding/Math/Primality Test/Fermat Method/SrcCode.cpp

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+
+#include <bits/stdc++.h>
+using namespace std;
+ 
+/* Iterative Function to calculate (a^n)%p in O(logy) */
+int power(int a, unsigned int n, int p)
+{
+    int res = 1;      // Initialize result
+    a = a % p;  // Update 'a' if 'a' >= p
+ 
+    while (n > 0)
+    {
+        // If n is odd, multiply 'a' with result
+        if (n & 1)
+            res = (res*a) % p;
+ 
+        // n must be even now
+        n = n>>1; // n = n/2
+        a = (a*a) % p;
+    }
+    return res;
+}
+ 
+// If n is prime, then always returns true, If n is
+// composite than returns false with high probability
+// Higher value of k increases probability of correct result.
+
+bool isPrime(unsigned int n, int k)
+{
+   // Corner cases
+   if (n <= 1 || n == 4)  return false;
+   if (n <= 3) return true;
+ 
+   // Try k times
+   while (k>0)
+   {
+       // Pick a random number in [2..n-2]        
+       // Above corner cases make sure that n > 4
+       int a = 2 + rand()%(n-4);  
+ 
+       // Fermat's little theorem
+       if (power(a, n-1, n) != 1)
+          return false;
+ 
+       k--;
+    }
+ 
+    return true;
+}
+ 
+// Driver Program
+int main()
+{
+    int k = 3;
+    isPrime(11, k)?  cout << " true\n": cout << " false\n";
+    isPrime(15, k)?  cout << " true\n": cout << " false\n";
+    return 0;
+}
+
+
+Output:
+
+true
+false

Competitive Coding/Math/Primality Test/Optimized School Method/README.md

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+Given a positive integer, check if the number is prime or not. A prime is a natural number greater than 1 that has no 
+positive divisors other than 1 and itself.
+
+Examples of first few prime numbers are {2, 3, 5, 7,...}
+
+Examples:
+
+**Input:**  n = 11  
+
+**Output:** true
+
+**Input:**  n = 15
+
+**Output:** false
+
+**Input:**  n = 1
+
+**Output:** false
+
+A simple solution is to iterate through all numbers from 2 to n-1 and for every number check if it divides n. If we find 
+any number that divides, we return false.   Instead of checking till n,  we can check till √n because a larger factor of n 
+must be a multiple of smaller factor that has been already checked.  The algorithm can be improved further by observing 
+that all primes are of the form 6k ± 1, with the exception of 2 and 3.  This is because all integers can  be  expressed 
+as (6k + i) for some integer k and for i = ?1, 0, 1, 2, 3, or 4; 2 divides  (6k + 0),  (6k + 2),  (6k + 4); and 3 divides 
+(6k + 3).  So a more efficient method is to test if n is divisible by 2 or 3,  then to check through all the numbers of 
+form 6k ± 1.
+
+Time complexity of this solution is O(root(n)).

Competitive Coding/Math/Primality Test/Optimized School Method/SrcCode.cpp

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+// A optimized school method based C++ program to check if a number is prime or not.
+
+#include <bits/stdc++.h>
+using namespace std;
+ 
+bool isPrime(int n)
+{
+    // Corner cases
+    if (n <= 1)  return false;
+    if (n <= 3)  return true;
+ 
+    // This is checked so that we can skip 
+    // middle five numbers in below loop
+    if (n%2 == 0 || n%3 == 0) return false;
+ 
+    for (int i=5; i*i<=n; i=i+6)
+        if (n%i == 0 || n%(i+2) == 0)
+           return false;
+ 
+    return true;
+}
+ 
+ 
+// Driver Program
+int main()
+{
+    isPrime(23)?  cout << " true\n": cout << " false\n";//use of ternary operator
+    isPrime(35)?  cout << " true\n": cout << " false\n";
+    return 0;
+}
+
+Output:
+
+true
+false