Liar Numbers - Numberphile

TL;DR

Using Fermat's Little Theorem to test if a number is prime leads to Carmichael numbers and better tests.

Transcript

DR GRIME: Let's have a look at one of these tests that they have, that they apply to see if a number is prime or not. Well we're going to use a fact I have mentioned before when talking about primes and we're going to use Fermat's Little Theorem. It's a theorem about prime numbers and what it says is that if you have a number which I'll call 'a'... Read More

Key Insights

• #️⃣ Fermat's Little Theorem assists in testing for prime numbers by relying on divisibility properties.
• 🪡 Carmichael numbers highlight the limitations of Fermat's method and the need for better prime testing techniques.
• 🏆 The Baillie-PSW test offers enhanced accuracy in prime testing by combining multiple tests.

Questions & Answers

Q: How does Fermat's Little Theorem help test for prime numbers?

Fermat's Little Theorem states that if a number raised to a prime minus the original number is divisible by the prime, the number is prime, aiding in prime number testing.

Q: What are Fermat liars and witnesses in the context of prime testing?

Fermat liars are composite numbers that falsely pass prime tests, while Fermat witnesses are numbers that correctly fail the test, indicating composite status.

Q: What are Carmichael numbers and why are they significant in prime testing?

Carmichael numbers, like 561, pass all prime tests, showing the limitations of Fermat's Little Theorem and the need for improved prime testing methods.

Q: How does the Baillie-PSW test improve upon Fermat's Little Theorem for prime testing?

The Baillie-PSW test combines two tests to enhance prime testing accuracy, with no counterexamples found so far, marking a significant improvement over Fermat's method.

Summary & Key Takeaways

• Fermat's Little Theorem states that if a number raised to a prime number minus the original number is divisible by the prime, the number is prime.

• Prime numbers pass this test, while composite numbers can fail, leading to Fermat liars and witnesses.

• Carmichael numbers, like 561, pass all prime tests, highlighting the need for better prime testing methods.