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amcavoy
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I just have one question about the proof. Why does [itex](p-1)!a^{p-1}=(p-1)!\pmod{p}[/itex]? It seems like it would be true if the (mod p) was instead (mod a).
Thanks for your help.
Thanks for your help.
Since a was invertible, they are all distinct mod p. So we have p - 1 numbers which are distinct mod p, so they must be congruent to 1, 2, 3, ..., p - 1 in some order.
apmcavoy said:It seems like it would be true if the (mod p) was instead (mod a).
"Proof of Fermat's Little Theorem" is a mathematical proof that was first stated by French mathematician Pierre de Fermat in 1640. It states that for any prime number p and any integer a, ap will always be congruent to a modulo p.
"Proof of Fermat's Little Theorem" is important because it has many applications in number theory and cryptography. It is also a fundamental theorem in algebra that helps to understand the properties of prime numbers.
Fermat's Little Theorem is used in cryptography to generate strong pseudorandom numbers, which are essential for ensuring the security of encryption algorithms. It is also used in the RSA algorithm, which is widely used in secure communication over the internet.
A congruence modulo p means that two numbers have the same remainder when divided by p. In other words, if a and b are two integers and p is a prime number, then a is congruent to b modulo p if and only if a mod p = b mod p.
The proof of Fermat's Little Theorem involves using mathematical induction and the binomial theorem. It can also be proven using group theory and Euler's theorem. The complete proof was provided by Leonhard Euler in 1736.