数学专业外文翻译--欧拉定理和费马定理

Euler’s Theorem and Fermat ’s Theorem

Book: Elementary Methods in number theory

Author :Melvyn B. Nathanson

Page :7167P P -

2.5 Euler’s Theorem and Fermat ’s Theorem

Euler ’s theorem and its corollary ,Fermat ’s theorem ,are fundamental results in number theory ,with many applications in mathematics and computer science .In the following sections we shall see how the Euler and Fermat theorems can be used to determine whether an integer is prime or composite ,and how they are applied in cryptography.

Theorem2.12（Euler ）Let m be a positive integer, and let a be an integer relatively prime to m .Then

()()m a m mod 1≡ϕ.

Proof. Let (){}

m r r ϕ ,1be a reduced set of residues modulo m .Since ()1,=m a ，we have ()()()m i m ar i ϕ ,11,== for 1,,()i m ϕ= .Consequently, for every (){}m i ϕ ,1∈there exists ()(){}m i ϕσ ,1∈such that

()()m r ar i i mod σ≡.

Moreover ，()m ar ar j i mod ≡ if and only if j i =，and so σ is a permutation of the set (){}m ϕ ,1 and (){}

m ar ar ϕ ,1 is also a reduced set of residues modulo m .It follows that ()()()()()()()m ar ar ar a m r r r m m mod 2121ϕϕϕ ≡

()()()()m r r r m m o d 21σσσ ≡

()()m r r r m mod 21ϕ ≡

Dividing by ()m r r r ϕ 21，we obtain

()()m a m mod 1≡ϕ

This completes the proof.

The following corollary is sometimes called Fermat ’s litter theorem.

Theorem 2.13 (Fermat ) Let p be a prime number .If the integer a is not divisible by p ,then

()p a r mod 11≡-

Moreover,

()p a a p mod ≡

for every integer a .

Proof . If p is prime and does not divide a, then ()1,=p a ，()1-=p p ϕ，and

()()p a a p p mod 11≡≡-ϕ

by Euler’s theorem. Multiplying this congruence by a ，we obtain

()p a a p mod ≡

If p divides a ,then this congruence also holds for a .

Let m be a positive integer and let a be an integer that is relatively prime to m .By Euler ’s theorem,()()m a m mod 1≡ϕ.The order of a with respect to the modulus m is the smallest positive integer d such that ()m a d mod 1≡.Then ()m d ϕ≤≤1.We shall prove that ()a ord m divides ()m ϕ for every integer a relatively prime to p

Theorem 2.14 Let m be a positive integer and a an integer relatively prime to m .If d is the order of a modulo m ,then ()m a a l

k mod ≡ if and only if ()d l k mod ≡.In particular, ()m a n mod 1≡ if and only if d divides n ,and so d divides ()m ϕ.

Proof. Since a has order modulo m ,we have ()m a d

mod 1≡.If ()d l k mod ≡，then dq l k +=，and so

()()m a a a a a l q d l dq l k mod ≡==+.

Conversely, suppose that ()m a a l k mod ≡.By the division algorithm, there exist integers q and

r such that

r dq l k +=-and 10-≤≤d r .

Then

()()m a a a a a a a r k r q d l r dq l k mod ≡==++