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 :
And for every prime factor q of (n-1),
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 % 9 = 256 % 9 = 4 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
C++
// C++ 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; primeFactors(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; } |
Python3
# Python3 program for Lucas Primality Test import random import math # Function to generate prime factors of n def primeFactors(n, factors): # If 2 is a factor if (n % 2 = = 0 ): factors.append( 2 ) while (n % 2 = = 0 ): n = n / / 2 # If prime > 2 is factor for i in range ( 3 , int (math.sqrt(n)) + 1 , 2 ): if (n % i = = 0 ): factors.append(i) while (n % i = = 0 ): n = n / / i if (n > 2 ): factors.append(n) return factors # This function produces power modulo # some number. It can be optimized to # using def power(n, r, q): total = n for i in range ( 1 , r): total = (total * n) % q return total def lucasTest(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 factors = [] factors = primeFactors(n - 1 , factors) # Array for random generator. This array # is to ensure one number is generated # only once rand = [i + 2 for i in range (n - 3 )] # Shuffle random array to produce randomness random.shuffle(rand) # Now one by one perform Lucas Primality # Test on random numbers generated. for i in range (n - 2 ): a = rand[i] if (power(a, n - 1 , n) ! = 1 ): return "composite" # This is to check if every factor # of n-1 satisfy the condition flag = True for k in range ( len (factors)): # 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 if __name__ = = "__main__" : print ( str ( 7 ) + " is " + lucasTest( 7 )) print ( str ( 9 ) + " is " + lucasTest( 9 )) print ( str ( 37 ) + " is " + lucasTest( 37 )) # This code is contributed by rutvik_56 |
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:
- Primality Test | Set 1 (Introduction and School Method)
- Primality Test | Set 2 (Fermat Method)
- Primality Test | Set 3 (Miller–Rabin)
- Primality Test | Set 4 (Solovay-Strassen)
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