Understanding Fermat's Little Theorem and Prime Testing

Dr. Grime explains Fermat's Little Theorem and its use in testing if numbers are prime. He demonstrates the theorem with examples, highlighting the concepts of Fermat liars and witnesses. The discussion then shifts to Carmichael numbers, which can pass primality tests despite being composite. The effectiveness of these tests is evaluated, and better alternatives like the Baillie-PSW test are introduced. The video concludes with a mention of advanced primality tests and the challenge of finding counterexamples.

What does Fermat's Little Theorem state about prime numbers?

• A number raised to a prime power minus the number itself is divisible by the prime.

• A number is prime if it passes the Baillie-PSW test.

• A number is prime if it is only divisible by 1 and itself.

• A number is composite if it can be divided by any number other than 1 and itself.

In the example using the number 5, which of the following calculations is NOT divisible by 5?

• 2⁵ - 2

• 1⁵ - 1

• 4⁵ - 4

• 3⁵ - 3

What is a Fermat liar?

• A number that is actually prime but fails Fermat's test.

• A composite number that passes Fermat's test for a specific base.

• A prime number that passes Fermat's test for all bases.

• A composite number that fails Fermat's test for all bases.

What is a Fermat witness?

• A number that fails Fermat's test for all bases.

• A number that passes Fermat's test for all bases.

• A number that shows a composite number is not prime.

• A number that proves a composite number is actually prime.

What is unique about Carmichael numbers?

• They are prime numbers that fail Fermat's test.

• They are composite numbers that pass Fermat's test for all bases.

• They are numbers that pass the Baillie-PSW test.

• They are numbers that are divisible by 1 and themselves only.

Which of the following is the first Carmichael number?

• 341

• 25

• 1729

• 561

What is the main limitation of Fermat's test as discussed in the video?

• It requires complex calculations.

• It can incorrectly identify composite numbers as prime.

• It is too slow for large numbers.

• It cannot identify any prime numbers.

What is the Baillie-PSW test?

• A test that uses two methods in conjunction to verify primality.

• A test that only works for numbers less than 1000.

• A test that is slower than Fermat's test.

• A test that has many known counterexamples.

What was the prize originally offered for finding a counterexample to the Baillie-PSW test?

• $500

• $30

• $100

• $620

What significant development in prime testing was published in 2002?

• A method that is 100% accurate and fast.

• A method that only works for small numbers.

• A method that is slower than previous tests.

• A method that has many counterexamples.