Prime Numbers

3.6 Lucas pseudoprimes 149
gcd(n, 2ab∆) = 1. This algorithm returns “n is a Lucas probable prime with
parameters a, b” if either n is prime or n is a Lucas pseudoprime with respect to
x 2 − ax + b. Otherwise, it returns “n is composite.”
1. [Auxiliary parameters]
A = a 2 b −1 − 2modn;
m = n −
∆
n /2;
2. [Binary Lucas chain]
Using Algorithm 3.6.7 calculate the last two terms of the sequence
(V0,V1,...,Vm,Vm+1), with initial values (V0,V1) =(2,A) and specific
rules V2j = V 2
j − 2modn and V2j+1 = VjVj+1 − A mod n;
3. [Declaration]
if(AVm ≡ 2Vm+1 (mod n)) return “n is a Lucas probable prime with
parameters a, b”;
return “n is composite”;
The algorithm for the Frobenius probable prime test is the same except that
Step [Declaration] is changed to
3’. [Lucas test]
if(AVm ≡ 2Vm+1) return “n is composite”;
and a new step is added:
4. [Frobenius test]
B = b (n−1)/2 mod n;
if(BVm ≡ 2(modn)) return “n is a Frobenius probable prime with
parameters a, b”;
return “n is composite”;
3.6.4 Theoretical considerations and stronger tests
If x 2 − ax + b is irreducible over Z and is not x 2 ± x + 1, then the Lucas
pseudoprimes with respect to x 2 − ax + b are rare compared with the primes
(see Exercise 3.26 for why we exclude x 2 ± x + 1). This result is in [Baillie and
Wagstaff 1980]. The best result in this direction is in [Gordon and Pomerance
1991]. Since the Frobenius pseudoprimes with respect to x 2 − ax + b are a
subset of the Lucas pseudoprimes with respect to this polynomial, they are if
anything rarer still.
It has been proved that for each irreducible polynomial x 2 − ax + b there
are infinitely many Lucas pseudoprimes, and in fact, infinitely many Frobenius
pseudoprimes. This was done in the case of Fibonacci pseudoprimes in [Lehmer
1964], in the general case for Lucas pseudoprimes in [Erdős et al. 1988], and
in the case of Frobenius pseudoprimes in [Grantham 2001]. Grantham’s proof
on the infinitude of Frobenius pseudoprimes works only in the case
∆
n =1.
There are some specific quadratics, for example, the polynomial x2 − x − 1for
the Fibonacci recurrence, for which we know that there are infinitely many
Frobenius pseudoprimes with
∆
n = −1 (see [Parberry 1970] and [Rotkiewicz