Prime Numbers

150 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
1973]). Recently, Rotkiewicz proved that for any x 2 − ax + b with ∆ = a 2 − 4b
not a square, there are infinitely many Lucas pseudoprimes n with
∆
n = −1.
In analogy to strong pseudoprimes (see Section 3.5), we may have strong
Lucas pseudoprimes and strong Frobenius pseudoprimes. Suppose n is an odd
prime not dividing b∆. In the ring R = Zn[x]/(f(x)) it is possible (in the case
∆
2 2 =1)tohavez= 1 and z = ±1. For example, take f(x) =x − x − 1,
n
n = 11, z =3+5x. However, if (x(a − x) −1 ) 2m = 1, then a simple calculation
(see Exercise 3.30) shows that we must have (x(a − x) −1 ) m = ±1. We have
from (3.10) and (3.11) that (x(a − x) −1 ) n−(∆ n) =1inR. Thus, if we write
n −
∆ s
n =2 t,wheretis odd, then
either (x(a − x) −1 ) t ≡ 1(mod(f(x),n))
or (x(a − x) −1 ) 2i t ≡−1(mod(f(x),n)) for some i, 0 ≤ i ≤ s − 1.
This then implies that
either Ut ≡ 0(modn)
or V 2 i t ≡ 0(modn) for some i, 0 ≤ i ≤ s − 1.
If this last statement holds for an odd composite number n coprime to b∆,
we say that n is a strong Lucas pseudoprime with respect to x 2 − ax + b. Itis
easy to see that every strong Lucas pseudoprime with respect to x 2 − ax + b
is also a Lucas pseudoprime with respect to this polynomial.
In [Grantham 2001] a strong Frobenius pseudoprime test is developed,
not only for quadratic polynomials, but for all polynomials. We describe the
quadratic case for
∆
2 S
n = −1. Say n − 1=2 T ,wherenis an odd prime not
dividing b∆ andwhere
∆
n
n = −1. From (3.10) and (3.11), we have x 2 −1 ≡ 1
(mod n), so that
either x T ≡ 1(modn)
or x 2i T ≡−1(modn) for some i, 0 ≤ i ≤ S − 1.
If this holds for a Frobenius pseudoprime n with respect to x 2 − ax + b,
we say that n is a strong Frobenius pseudoprime with respect to x 2 − ax + b.
(That is, the above congruence does not appear to imply that n is a Frobenius
pseudoprime, so this condition is put into the definition of a strong Frobenius
pseudoprime.) It is shown in [Grantham 1998] that a strong Frobenius
pseudoprime n with respect to x2 − ax + b, with
∆
n = −1, is also a strong
Lucas pseudoprime with respect to this polynomial.
As with the ordinary Lucas test, the strong Lucas test may be
accomplished in time bounded by the cost of two ordinary pseudoprime
tests. It is shown in [Grantham 1998] that the strong Frobenius test may
be accomplished in time bounded by the cost of three ordinary pseudoprime
tests. The interest in strong Frobenius pseudoprimes comes from the following
result from [Grantham 1998]: