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matqkks
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What use are Fermat’s Little Theorem and Wilson’s theorems in number theory? Do these theorems have any real life applications? We cannot use them to find primes as both are pretty inefficient for large numbers.
Fermat's Little Theorem is a mathematical theorem named after French mathematician Pierre de Fermat. It states that if a is a positive integer and p is a prime number, then a^p - a is a multiple of p.
Fermat's Little Theorem is often used in number theory and cryptography. It is also used in the fields of algebraic geometry and algebraic number theory.
Fermat's Little Theorem is significant because it provides a simple and efficient way to check whether a number is prime. It is also used to prove other theorems, such as Euler's theorem and Wilson's theorem.
Fermat's Little Theorem only applies to prime numbers, so it cannot be used to prove the primality of composite numbers. It is also not a foolproof method for determining whether a number is prime, as there are rare cases where a composite number can satisfy the theorem.
Yes, there are extensions of Fermat's Little Theorem that apply to non-prime moduli. One example is Euler's theorem, which states that if a and n are coprime positive integers, then a^(φ(n)) ≡ 1 (mod n), where φ(n) is the Euler totient function.