Lucas Primality Test

A number p greater than one is prime if and only if the only divisors of p are 1 and p. First few prime numbers are 2, 3, 5, 7, 11, 13, …

The Lucas test is a primality test for a natural number n, it can test primality of any kind of number.

It follows from Fermat’s Little Theorem: If p is prime and a is an integer, then a^p is congruent to a (mod p )

Lucas’ Test : A positive number n 
is prime if there exists an integer a (1 < a 
< n) such that :

a^{{n-1}}\ \equiv \ 1{\pmod n}

And for every prime factor q of (n-1),

a^{{({n-1})/q}}\ \not \equiv \ 1{\pmod n}

Examples :

Input :  n = 7
Output : 7 is Prime
Explanation : let's take a = 3, 
then 3^6 % 7 = 729 % 7 = 1 (1st 
condition satisfied). Prime factors 
of 6 are 2 and 3,
3^(6/2) % 7 = 3^3 % 7 = 27 % 7 = 6
3^(6/3) % 7 = 3^2 % 7 = 9 % 7 = 2
Hence, 7 is Prime  

Input :  n = 9
Output : 9 is composite
Explanation : Let's take a = 2,
then 2^8 % 8 = 256 % 8 = 0
Hence 9 is composite 

lucasTest(n):
If n is even
    return composite
Else
   Find all prime factors of n-1
   for i=2 to n-1
      pick 'a' randomly in range [2, n-1]
      if a^(n-1) % n not equal 1:
          return composite
      else 
          // for all q, prime factors of (n-1)
          if a^(n-1)/q % n not equal 1 
             return prime
   Return probably prime

Problems Associated with Lucas’s test are :

  • Knowing all of the prime factors of n-1
  • Finding an appropriate choice for a
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      // CPP Program for Lucas Primality Test
      #include <bits/stdc++.h>
      using namespace std;
        
      // function to generate prime factors of n
      void primeFactors(int n, vector<int>& factors)
      {
          // if 2 is a factor
          if (n % 2 == 0)
              factors.push_back(2);
          while (n % 2 == 0)
              n = n / 2;
                
          // if prime > 2 is factor
          for (int i = 3; i <= sqrt(n); i += 2) {
              if (n % i == 0)
                  factors.push_back(i);
              while (n % i == 0)
                  n = n / i;
          }
          if (n > 2)
            factors.push_back(n);
      }
        
      // this function produces power modulo 
      // some number. It can be optimized to 
      // using 
      int power(int n, int r, int q)
      {
          int total = n;
          for (int i = 1; i < r; i++)
              total = (total * n) % q;
          return total;
      }
        
      string lucasTest(int n)
      {
          // Base cases 
          if (n == 1)
              return "neither prime nor composite";
          if (n == 2)
              return "prime";
          if (n % 2 == 0)
              return "composite1";
                
                
          // Generating and storing factors 
          // of n-1
          vector<int> factors;
          prime_factors(n - 1, factors);
        
          // Array for random generator. This array 
          // is to ensure one number is generated 
          // only once
          int random[n - 3];
          for (int i = 0; i < n - 2; i++)
              random[i] = i + 2;
                
          // shuffle random array to produce randomness
          shuffle(random, random + n - 3, 
                     default_random_engine(time(0)));
        
          // Now one by one perform Lucas Primality
          // Test on random numbers generated.
          for (int i = 0; i < n - 2; i++) {
              int a = random[i]; 
              if (power(a, n - 1, n) != 1) 
                  return "composite";
        
              // this is to check if every factor 
              // of n-1 satisfy the condition
              bool flag = true;
              for (int k = 0; k < factors.size(); k++) {
                  // if a^((n-1)/q) equal 1
                  if (power(a, (n - 1) / factors[k], n) == 1) {
                      flag = false;
                      break;
                  }
              }
        
              // if all condition satisfy
              if (flag)
                  return "prime";
          }
          return "probably composite";
      }
        
      // Driver code
      int main()
      {
          cout << 7 << " is " << lucasTest(7) << endl;
          cout << 9 << " is " << lucasTest(9) << endl;
          cout << 37 << " is " << lucasTest(37) << endl;
          return 0;
      }

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      Output:
      

      7 is prime
9 is composite
37 is prime

      

      This method is quite complicated and inefficient as compared to other primality tests. And the main problems are factors of ‘n-1’ and choosing appropriate ‘a’. 
      

      Other Primality tests:
      


      This article is contributed by Shubham Rana. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
      

      Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
      

      
          
          
          
          
      
            
      
    
      
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