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 : 

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

Output:

7 is prime
9 is composite
37 is prime

Time Complexity: O(nlogn)

Auxiliary Space: O(n)

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)