MAT 3930
Primality Test
Theorem: (Fermat’s Little Theorem, modified version) Let p be a prime and let a be a
number with gcd(a, p) = 1. Then
ap ≡ a (mod p)
Theorem: (Contrapositive to FlT) Let n be a positive number and let a be a number with
gcd(a, n) = 1. If
an 6≡ a (mod n)
then a is composite.
Definition: A composite number n is called a pseudoprime base a if n satisfies
an ≡ a (mod n)
Definition: A composite number n is called a Charmichael number if it is a pseudoprime
with respect to every base a.
Definition: Let n be an odd composite number and write n − 1 = 2k q, where q is odd. An
integer a satisfying gcd(n, a) = 1 is called a witness for n if both of the following hold:
(i) aq 6≡ 1 (mod n)
i
(ii) a2 q 6≡ −1 (mod n) for 0 ≤ i ≤ k − 1
Theorem: If a is a witness for n, then n is composite.
Definition: If a is not a witness for n, then we say n is a strong pseudoprime base a.
Theorem: For any odd composite number n, at least 75% of the numbers less than n are
witnesses for n. In particular, there are no strong Charmichael numbers.
Testing for Primality: Given an odd integer n, check to see if a is a witness for all numbers
from 2 to 50. If any of these numbers are witnesses for n, then n is composite. If none of
these numbers are witnesses, then there is less than a 1 in 2100 chance that the number is
prime. We say that n is a probable prime, and most people in the world would be willing to
bet that n is prime.