- #1
micromass said:
stringy said:
Fermat's Little Theorem is a mathematical theorem named after French mathematician Pierre de Fermat. It states that if p is a prime number, then for any integer a, the number a^p - a is an integer multiple of p.
Fermat's Little Theorem is used in cryptography to create public key systems. In particular, it is used in the creation of the RSA algorithm, which is a widely used method for secure communication over the internet.
No, Fermat's Little Theorem cannot be used to definitively determine if a number is prime. However, it can be used as a primality test, meaning that if the theorem holds true for a specific number, it is very likely to be prime. However, there are rare cases where the theorem holds true for non-prime numbers.
The proof of Fermat's Little Theorem is quite complex and involves concepts from number theory and abstract algebra. It was originally proved by Leonhard Euler in 1736, and there have been several other proofs since then. A simplified version of the proof can be found in most high school or college-level mathematics textbooks.
Yes, Fermat's Little Theorem can be extended to non-prime moduli. This is known as Euler's generalization of Fermat's Little Theorem, and it states that if n is any positive integer and a is an integer relatively prime to n, then a^(φ(n)) ≡ 1 (mod n), where φ(n) is Euler's totient function.
