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