Practice Example Of A Carmichael Number (561) (12.5.1) - Introduction to Fermat’s Little Theorem and Primality Testing

Practice: Example of a Carmichael Number (561)

Lesson 12.5.1Introduction to Fermat’s Little Theorem and Primality Testing

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6 sample questions

Question 1.
State Fermat's Little Theorem.
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Think about the relation between primes and modular arithmetic.
Question 2.
What is an example of a Carmichael number?
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Recall that it misleads primality tests while being composite.
3Question 3.
What does Fermat's Little Theorem state?
- If p is prime
- then a^p ≡ 0 (mod p)
- If p is prime
- then a^(p-1) ≡ 1 (mod p)
- If a is even
- a^2 is even
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Remember the key formula in the theorem.
4Question 4.
Is 561 a prime number?
- True
- False
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Recollect the definition of prime numbers.
Question 5.
Prove that if a number n satisfies b^(n-1) ≡ 1 for all bases b co-prime to n, then n is a Carmichael number.
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Use the prime factorization of n and analyze the implications for all bases.
Question 6.
In a practical application, explain how one could modify a basic primality test to take Carmichael numbers into account.
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Consider the diversity of bases used in testing.

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