Practice Primality Testing Algorithm Limitations (12.3.2) - Introduction to Fermat’s Little Theorem and Primality Testing

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6 sample questions

Question 1.
What is Fermat's Little Theorem?
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Think about how it relates to primes and modular arithmetic.
Question 2.
Explain what a pseudo prime is.
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Consider an example of a composite number.
3Question 3.
What does Fermat's Little Theorem state for a prime p?
- a^(p-1) ≡ 1 (mod p)
- a^(p) ≡ 1 (mod p)
- p | a
- gcd(a
- p) = 0
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Recall the theorem definition directly.
4Question 4.
Is 341 a pseudo prime?
- True
- False
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Think about its factors.
Question 5.
Prove that 561 is a Carmichael number by demonstrating it satisfies Fermat's theorem for all valid bases.
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Tackle it by considering GCD conditions for 3, 11, and 17, and apply Fermat's theorem.
Question 6.
Find a composite number greater than 100 which is not a Carmichael number and explain why it fails the Fermat test.
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Test various bases and demonstrate a failure in at least one case.

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