Prime Numbers

3.6 Lucas pseudoprimes 143
The Fibonacci pseudoprime test is not just a curiosity. As we shall see
below, it can be implemented on very large numbers. In fact, it takes only
about twice as long to run a Fibonacci pseudoprime test as a conventional
pseudoprime test. And for those composites that are ±2 (mod5)itis,when
combined with the ordinary base-2 pseudoprime test, very effective. In fact, we
know no number n ≡±2 (mod 5) that is simultaneously a base-2 pseudoprime
and a Fibonacci pseudoprime; see Exercise 3.41.
In proving Theorem 3.6.1 it turns out that with no extra work we
can establish a more general result. The Fibonacci sequence satisfies the
recurrence uj = uj−1 + uj−2, with recurrence polynomial x 2 − x − 1. We shall
consider the more general case of binary recurrent sequences with polynomial
f(x) =x 2 − ax + b, wherea, b are integers with ∆ = a 2 − 4b not a square. Let
Uj = Uj(a, b) = xj − (a − x) j
x − (a − x)
(mod f(x)),
Vj = Vj(a, b) =x j +(a − x) j (mod f(x)), (3.8)
where the notation means that we take the remainder in Z[x] upon division by
f(x). The sequences (Uj), (Vj) both satisfy the recurrence for the polynomial
x 2 − ax + b, namely,
Uj = aUj−1 − bUj−2, Vj = aVj−1 − bVj−2,
and from (3.8) we may read off the initial values
U0 =0, U1 =1, V0 =2, V1 = a.
If it was not already evident from (3.8), it is now clear that (Uj), (Vj) are
integer sequences.
In analogy to Theorem 3.6.1 we have the following result. In fact, we can
read off Theorem 3.6.1 as the special case corresponding to a =1,b = −1.
Theorem 3.6.3. Let a, b, ∆ be as above and define the sequences (Uj), (Vj)
via (3.8). If p is a prime with gcd(p, 2b∆) = 1, then
U ∆ p−( ≡ 0(modp). (3.9)
p)
Note that for ∆ = 5 and p odd,
5 p
p = 5 , so the remark following Theorem
3.6.1 is justified. Since the Jacobi symbol
∆
n (see Definition 2.3.3) is equal
to the Legendre symbol when n is an odd prime, we may turn Theorem 3.6.3
into a pseudoprime test.
Definition 3.6.4. We say that a composite number n with gcd(n, 2b∆) = 1
is a Lucas pseudoprime with respect to x2 − ax + b if U ∆ n−( ≡ 0(modn).
n)
Since the sequence (Uj) is constructed by reducing polynomials modulo
x 2 − ax + b, and since Theorem 3.6.3 and Definition 3.6.4 refer to this
sequence reduced modulo n, we are really dealing with objects in the ring