12.4.2. Characteristics of Carmichael Numbers

Try these first

1. 1.

   What is a Carmichael number?

   Hint

   Recall the definition we covered.

2. 2.

   Give an example of a Carmichael number.

   Hint

   Think of the examples discussed in class.

3. 3.

   What is a Carmichael number?

   • A prime number
   • A composite number satisfying Fermat's theorem for all coprime bases
   • A pseudo prime for certain bases

   Hint

   Remember the defining characteristics we reviewed.

4. 4.

   True or False: All composite numbers are Carmichael numbers.

   • True
   • False

   Hint

   Consider the special properties of Carmichael numbers.

5. 5.

   Identify and verify if 1729 is a Carmichael number and discuss its properties.

   Hint

   Check its prime factors: 7, 13, and 19, ensuring all conditions are satisfied.

6. 6.

   Create an algorithm or step-by-step method that incorporates checks for Carmichael numbers in primality testing.

   Hint

   Use the characteristics we discussed to create conditions.