dissertacao.pdf

So if we use contrapositive of this theorem, we get a test to check whether a
given number is composite:
Theorem 8. An odd positive integer N is composite if there exists a positive
integer a such that (a, N) = 1 and
a N−1 ≇ 1 (mod N) (27)
There are composite numbers N that satisfy a N−1 ≡ 1 (mod N) for some a
in the conditions of the theorem. In this case, N is called a base-a pseudoprime.
We call it pseudoprime because, if we run Fermat’s Primality Test (FPT) for
N with such a base a, the test will not identify N as composite. One way to
round this problem is to simply run the test again with a different a : base-a
pseudoprime form quite different sets for different values of a, so we have the
hope of identifying N as a composite by simply trying out different values of
a. There is, however, one problem: there are composite numbers N which are
base-a pseudoprime for all a satisfying the conditions of the theorem. These
numbers are called Carmichael Numbers.
Definition 15. A composite number N is called a Carmichael Number if it
satisfies:
for every positive integer a such that (a, N) = 1.
a N−1 ∼ = 1 (mod N) (28)
So a Carmichael number will never be identified as composite by FPT. Al-
though they are highly infrequent, Carmichael numbers are an infinite set so
they represent the biggest flaw of FPT. For this reason, FTP alone is not a
deterministic test.
22