Lucas-Lehmer primality test
The Lucas-Lehmer primality test
is a method of testing the primality
of some number n
based on testing whether some other number is primitive
If there exists some a less than n and greater than 1 such that firstly an-1≡1 and then
represents the prime factors of n
-1, then n
is prime, since this is the requirement for a
to be primitive mod n
, resulting then the multiplicative order
to be n
For example, take n=71, n-1=70=(2)(5)(7).
Take a=2 first:
This doesn't show that the order of 2 mod 70 is 1 because some factor of 70 may also work above. So check 70's factors:
So 2 is primitive mod 71 and thus 71 is prime.
If the factors of n-1 are not easily obtained, this method becomes difficult to use as these factors must be obtained in the a(n-1)/qi terms.