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FermatPrimality

# FermatPrimality - The Fermat Primality Test is...

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(2) (1) (3) The Fermat Primality Test is probabilistic The Fermat Primality test is described in the text on p. 177. We start with an odd positive integer n and check to see if it is prime or composite. One chooses a value a with 1 < a < n - 1 and then uses modular exponentiation to compute mod n. If the result is different than 1, we can state categorically that n is composite, whereas if the result of the computation is 1, then we state that n is probably prime. That means that we can sometimes get a result of 1 even when n is composite. In particular, the lecture pointed out that we had already seen an example where absolutely every such value a has the property that mod n = 1, even though n is composite. The HW problem p. 105 #16 is an example of a Carmichael Number, and we had to show that every possible random choice of a would lead to a failure of the Fermat Primality Test, producing a value 1 for the composite number n = 1729.

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