12.4.1. Definition of Pseudoprimes

Try these first

1. 1.

   Define what a pseudoprime is.

   Hint

   Think about the related concept of primality testing.

2. 2.

   Give an example of a Carmichael number.

   Hint

   Recall that it satisfies specific conditions for all coprime bases.

3. 3.

   What defines a pseudoprime?

   • A prime number
   • A composite number satisfying Fermat's Theorem
   • A number that is always prime

   Hint

   Focus on its relation to Fermat’s theorem.

4. 4.

   True or False: All Carmichael numbers are pseudoprime.

   • True
   • False

   Hint

   Recall the definitions of both concepts.

5. 5.

   Prove that a number which is not a pseudoprime cannot pass primality tests for every base.

   Hint

   Consider analyzing the conditions laid out by Fermat's theorem as you reason through.

6. 6.

   Given a composite number, demonstrate how to identify if it is a Carmichael number.

   Hint

   Use the definition of Carmichael numbers to guide your reasoning.