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A short class with Sarah and her students. The other ways are under More.
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Let's explore Fermat's Little Theorem. It states that if p is a prime number and a is an integer not divisible by p, then a raised to the power of p-1 is congruent to 1 mod p. Can anyone share what this congruence means?
It means when you divide a^(p-1) by p, the remainder is 1!
Exactly! We often express that with congruence notation: a^(p-1) ≡ 1 mod p. This theorem helps us simplify calculations involving large powers. Can you think of why this might be useful?
It helps compute large powers without calculating them directly!
Right! That's an essential aspect in cryptography as well!
Now, what do you think is the corollary of Fermat's Little Theorem?
Is it that a^p ≡ a mod p for any integer a?
Exactly! It includes all integers a, not just those co-prime to p. Let’s break it into two cases: what happens if p divides a and if it does not.
If p divides a, the remainder is 0, so a^p ≡ a mod p holds true.
And if p does not divide a, we use Fermat's theorem to show a^p - a is zero.
Great observations! Hence, both cases confirm the corollary!
To prove Fermat's Little Theorem, we start by considering distinct multiples of a. Can anyone explain the contradicting argument used?
We assume two multiples of a give the same remainder when divided by p and show it leads us to a contradiction!
Correct! This contradiction is fundamental as it emphasizes the uniqueness of our multiples under modulo p.
Does it involve GCD properties as well?
It does! Because we need to ensure a is co-prime to p to apply Fermat's reasoning.
Fermat’s Little Theorem can help identify if a number n is prime. However, there's a catch. What happens if the condition fails?
We can declare n as composite right away!
Yes! But if bn - 1 ≡ 1 mod n holds, can we declare n as prime?
Not necessarily, as there are pseudo primes!
Exactly! Such cases lead to false conclusions in primality testing, requiring further checks.
Carmichael numbers are composite numbers that satisfy Fermat's theorem for every base. Can anyone give me an example of one?
561 is an example!
Yes! And they lead to failure in the primality test because they also satisfy bn - 1 ≡ 1 for all bases co-prime to n.
So, it's crucial to check beyond just Fermat's theorem to assert if a number is actually prime!
Precisely! It’s an important lesson in the reliability of primality tests.
If p is prime and a is co-prime to p, then a^(p-1) ≡ 1 mod p.
The relation a^p ≡ a mod p for any integer a.
Composite numbers that satisfy Fermat's theorem for at least one base.
Special composite numbers that satisfy Fermat’s theorem for all bases co-prime to the number.
When primes are near, and numbers cheer, Euler gets glory and Fermat's theorem is clear.
Imagine a castle (the prime number p) protected by the brave knight a. Only those knights who are strong (co-prime) can show that a can defend against all enemies raised to p-1.
To remember Fermat's theorem: 'A Knight, P-Power, Stands Strong' (A for a, P for prime, Stands for congruency).
F.L.T. stands for Fermat's Little Theorem.
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What is Fermat's Little Theorem?
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What is a pseudo prime?
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What is a Carmichael number?
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A theorem stating that if p is a prime number and a is an integer such that p does not divide a, then a^(p-1) ≡ 1 mod p.
A statement that follows with little or no additional proof from an already established statement.
A relation that expresses the equivalence of two integers with respect to a modulus.
A composite number that satisfies the condition of Fermat's Little Theorem for a given base.
A composite number that is a pseudo prime for every base co-prime to it.
Fermat's Little Theorem states that if a prime number p does not divide integer a, then a raised to p-1 is congruent to 1 mod p.
The section discusses Fermat's Little Theorem and its corollary, proofs, and applications in primality testing and the concept of Carmichael numbers, highlighting how they interact in number theory.
Fermat’s Little Theorem states that if p is a prime number and a is an integer where p does not divide a (i.e., a is co-prime to p), then:
This theorem is fundamental in number theory and is significant in various applications, particularly in primality testing. The theorem is distinguished from Fermat’s Last Theorem due to its simpler nature.
An important corollary is:
This applies for any integer a, whether or not it is co-prime to p. The proof follows two cases, either when a is divisible by p or not.
The proof builds upon the concept of distinct multiples of a and relies on a contradiction argument. For distinct integers r and s such that their multiples of a yield the same remainder when divided by p, we arrive at a contradiction under the assumption that p is a prime number.
Fermat's Little Theorem is used to check if n is prime by picking an integer b co-prime to n and checking if:
However, the mere satisfaction of this condition does not guarantee that n is prime, as illustrated by pseudo primes and Carmichael numbers.
Carmichael numbers are composite numbers that satisfy Fermat's theorem for all bases co-prime to them, thus failing the primality test. The section provides examples such as 561, which serves as a Carmichael number. The interplay between these concepts illustrates the complexities of primality testing in number theory.