Prime Numbers

3.5 Probable primes and witnesses 139
Lemma 3.5.8. Say n is an odd composite with n − 1=2 s t, t odd. Let ν(n)
denote the largest integer such that 2 ν(n) divides p−1 for each prime p dividing
n. Ifn is a strong pseudoprime base a, then a 2ν(n)−1 t ≡±1(modn).
Proof. If a t ≡ 1(modn), it is clear that the conclusion of the lemma holds.
Suppose we have a 2i t ≡−1(modn) andletp be a prime factor of n. Then
a 2i t ≡ −1(modp). If k is the order of a (mod p) (that is, k is the least
positive integer with a k ≡ 1(modp)), then k divides 2 i+1 t, but k does not
divide 2 i t. Thus the exact power of 2 in the prime factorization of k must be
2 i+1 .Butalsok divides p − 1, so that 2 i+1 |p − 1. Since this holds for each
prime p dividing n, wehavei +1≤ ν(n). Thus, a 2ν(n)−1 t ≡ 1(modn) or−1
(mod n) depending on whether i +1