Prime Numbers

178 Chapter 4 PRIMALITY PROVING
Since n is prime, this must be a trivial factorization of n, thatis,
c1 + tF −|u| =0,
which implies c4 = t. Butc4 ≥ F ≥ n 3/10 ≥ 214 3/10 > 5 ≥ t, a contradiction.
So if (1) fails, n must be composite. It is obvious that if n is prime, then (2)
holds. ✷
As with Theorem 4.1.5, if Theorem 4.1.6 is to be used as a primality
test, one should use Algorithm 9.2.11 as a subroutine to recognize squares. In
addition, one should use Newton’s method or a divide and conquer strategy
to search for integral roots of the cubic polynomial in condition (2) of the
theorem. We next embody Theorems 4.1.3-4.1.6 in one algorithm.
Algorithm 4.1.7 (The n − 1 test). Supposewehaveanintegern ≥ 214 and
that (4.2) holds with F ≥ n 3/10 . This probabilistic algorithm attempts to decide
whether n is prime (YES) or composite (NO).
1. [Pocklington test]
Choose random a ∈ [2,n− 2];
if(a n−1 ≡ 1(modn)) return NO; // n is composite.
for(prime q|F ) {
g =gcd (a (n−1)/q mod n) − 1,n ;
if(1