Prime Numbers

142 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
is a witness for n. In [Burthe 1997] it is shown that W (n) ≤ n c+o(1) as
n → ∞ through the odd composites, where c = 1/(6 √ e). Heath-Brown
(see [Balasubramanian and Nagaraj 1997]) has recently shown this with
c =1/10.82.
We close this section with the Miller primality test. It is based on
Theorem 3.5.12 and shows that if the extended Riemann hypothesis holds,
then primality can be decided in deterministic polynomial time.
Algorithm 3.5.13 (Miller primality test). We are given an odd number n>
1. This algorithm attempts to decide whether n is prime (YES) or composite (NO).
If NO is returned, then n is definitely composite. If YES is returned, n is either
prime or the extended Riemann hypothesis is false.
1. [Witness bound]
W =min{ 2ln 2 n ,n− 1};
2. [Strong probable prime tests]
for(2 ≤ a ≤ W ) {
Decide via Algorithm 3.5.2 whether n is a strong probable prime base a;
if(n is not a strong probable prime base a) return NO;
}
return YES;
3.6 Lucas pseudoprimes
We may generalize many of the ideas of the past two sections to incorporate
finite fields. Traditionally the concept of Lucas pseudoprimes has been cast
in the language of binary recurrent sequences. It is profitable to view this
pseudoprime construct using the language of finite fields, not just to be
fashionable, but because the ideas then seem less ad hoc, and one can
generalize easily to higher order fields.
3.6.1 Fibonacci and Lucas pseudoprimes
The sequence 0, 1, 1, 2, 3, 5,... of Fibonacci numbers, say uj is the j-th one
starting with j = 0, has an interesting rule for the appearance of prime factors.
Theorem 3.6.1. If n is prime, then
un−εn ≡ 0(modn), (3.7)
where εn =1when n ≡±1(mod5), εn = −1 when n ≡±2(mod5),and
εn =0when n ≡ 0(mod5).
Remark. The reader should recognize the function εn. It is the Legendre
symbol
n
5 ; see Definition 2.3.2.
Definition 3.6.2. We say that a composite number n is a Fibonacci
pseudoprime if (3.7) holds.
For example, the smallest Fibonacci pseudoprime coprime to 10 is 323.