Euler's function (1)

If you have the basics of section 1 and section 2 in number theory, your understanding of Euler's function is simpler,

The first section of number theory: the concept and basic nature of homophone

Section 2: residual class and complete residual series

First, we need to know several concepts:

Same definition: for example, X1 % m = r

X2 % m = R, then we will say that X1 and X2 are two numbers about M coolder

Theorem on the same remainder: If integers A and B are about M same remainder, there is a necessary condition: M | A-B, that is, a = B + MT; t is an integer.

Definition of the remainder class: Put all elements with the same Modulus m to the R (r <m) to form a set of numbers, and this set is called the remainder class, it is represented by KR ..

Definition of a completely residual system: the set of elements in all the remainder classes obtained by Modulo M is called the completely residual system of Modulo M. For example:

About 7, one of the completely residual systems is 0, 1, 2, 3, 4, 5, 6 can also be: 2, 3, 4, 5, 6, 8 and so on ..

Simplify the definition of the residual system: it is to retrieve and m in the completely residual SystemMutual QualityA set of numbers composed of elements...

There are still many properties and theorems that will not be listed...

Euler's function: Very simple,Simply put, the number of elements in the remainder is simplified...

For example, the complete residual series of Modulo 8: 1, 2, 3, 4, 5, 6, 7, 8

Simplified remaining systems: 1, 3, 5, and 7

There are four elements in the simplified residual series of Modulo 8, so E (8) = 4 ......

Note: E (n) = 1 <=> n = 1 or N = 2;

Small Conclusion 1: If P is a prime number, then E (P) = p-1, E (P ^ n) = P ^ N-P ^ (n-1 );

Proof: (1): Because p is a prime number, elements in the complete residue of model P include: 1, 2, 3, 4,... P-1;

After removing the multiples of one P, it becomes simplified to the following: 1, 2, 3, 4,... P-2, P-1.

Therefore, E (P) = p-1;

(2): In the same sense, the complete Remainder of the modulo p ^ N has the following elements: 1, 2 ,... p, p + 1 .... 2 * P, 2 * p + 1 .... P ^ (n-1) * P-1, P ^ (n-1) * P

Except for the multiples of P ^ (n-1) P, the remainder is simplified to 1, 2 ,... p-1, p + 1 ,... 2 p-1, 2 * p + 1 ....

Therefore, E (P ^ n) = P ^ N-P ^ (n-1 );

Conclusion 2: In a modulo m remainder class, if an element and m are mutually dependent, all elements in the class are mutually dependent ..

Proof: assume there is a m remainder class Kr = {x1, x2, X3, x4......}, and (x1, m) = 1;

Any element XN from KR;

Because XN is the same as X1, M | xn-X1 => xn = X1 + MT (T = 1, 2, 3 ,...)

And because (x1, m) = 1; so xn = MT + X1 => obtained by the method of Band Division: (Xn, m) = (M, X1) = 1;

So the conclusion is true...

Based on Conclusion 2, we can draw conclusion 3 ..

Conclusion 3: In a modulo m remainder class, if the maximum convention between an element and M is D, all elements in the remainder class and m are D ..

The proof method is the same as the proof of conclusion 2...