Prime Numbers

136 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
2. [Power of 2 in n − 1]
for(j ∈ [1,s− 1]) { // j is a dummy counter.
b = b 2 mod n;
if(b == n − 1) return “n is a strong probable prime base a”;
}
return “n is composite”;
This test was first suggested in [Artjuhov 1966/67], and a decade later,
J. Selfridge rediscovered the test and popularized it.
We now consider the possibility of showing that an odd number n is
composite by showing that (3.4) fails for a particular number a. For example,
we saw in the previous section that 341 is pseudoprime base 2. But (3.4) does
not hold for n = 341 and a =2.Indeed,wehave340=2 2 · 85, 2 85 ≡ 32
(mod 341), and 2 170 ≡ 1 (mod 341). In fact, we see that 32 is a nontrivial
square root of 1 (mod 341).
Now consider the pair n =91anda = 10. We have 90 = 2 1 · 45 and
10 45 ≡−1 (mod 91). So (3.4) holds.
Definition 3.5.3. We say that n is a strong pseudoprime base a if n is an
odd composite, n − 1=2 s t,witht odd, and (3.4) holds.
Thus, 341 is not a strong pseudoprime base 2, while 91 is a strong pseudoprime
base 10. J. Selfridge proposed using Theorem 3.5.1 as a pseudoprime test in
the early 1970s, and it was he who coined the term “strong pseudoprime.” It
is clear that if n is a strong pseudoprime base a, thenn is a pseudoprime base
a. The example with n = 341 and a = 2 shows that the converse is false.
For an odd composite integer n we shall let
S(n) ={a (mod n) :n is a strong pseudoprime base a}, (3.5)
and let S(n) =#S(n). The following theorem was proved independently in
[Monier 1980] and [Rabin 1980].
Theorem 3.5.4. For each odd composite integer n>9 we have S(n) ≤
1
4 ϕ(n).
Recall that ϕ(n) is Euler’s function evaluated at n. Itisthenumberof
integers in [1,n] coprime to n; that is, the order of the group Z∗ n. If we
know the prime factorization of n, it is easy to compute ϕ(n): We have
ϕ(n) =n
p|n
(1 − 1/p), where p runs over the prime factors of n.
Before we prove Theorem 3.5.4, we first indicate why it is a significant
result. If we have an odd number n and we wish to determine whether it
is prime or composite, we might try verifying (3.4) for some number a with
1