12.1.6. Primality Testing Algorithms

Try these first

1. 1.

   What does Fermat's Little Theorem state for prime p?

   Hint

   Think about the relationship between primes and modular arithmetic.

2. 2.

   Provide an example of a Carmichael number.

   Hint

   Carmichael numbers are always composite.

3. 3.

   According to Fermat's Little Theorem, if p is prime and a is coprime to p, what is true?

   • `a^p ≡ 0 mod p`
   • `a^(p-1) ≡ 1 mod p`
   • `p^a ≡ a mod p`

   Hint

   Recall the main statement of the theorem.

4. 4.

   Are Carmichael numbers always prime?

   • True
   • False

   Hint

   Think of their definition and characteristics.

5. 5.

   Prove that 341 is a Carmichael number using Fermat's test for a random base.

   Hint

   Pick bases carefully that are less than `341`.

6. 6.

   Create a detailed algorithm that efficiently uses Fermat's theorem to compute a^b mod p, explaining each step.

   Hint

   Think about employing the divide-and-conquer strategy!