12.5. Examples and Concluding Thoughts

Try these first

1. 1.

   What does Fermat's Little Theorem state?

   Hint

   Look at the conditions involving p and a.

2. 2.

   Define a Carmichael number.

   Hint

   Think of numbers that behave like primes under certain tests.

3. 3.

   What is the significance of Fermat's Little Theorem?

   • Proves all numbers are primes
   • Forms a basis for primality testing
   • Is irrelevant

   Hint

   Consider its applications in real-life situations.

4. 4.

   True or False: All composite numbers can be identified using Fermat's Little Theorem.

   • True
   • False

   Hint

   Think critically about the exceptions in number theory.

5. 5.

   Prove that 341 is a false prime using Fermat's theorem.

   Hint

   Use a base like 2 or 3, and calculate carefully.

6. 6.

   Describe an efficient way to test for Carmichael numbers and implement it in a function.

   Hint

   Look up conditions for known Carmichael numbers.