概念
![同餘方程](/img/1/596/wZwpmL2cDOxQTM3cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
定義1:設是整係數多項式,稱
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
是關於未知數的模的同餘方程,簡稱為模的同餘方程。
![同餘方程](/img/d/7ac/wZwpmL1YzM5AjN0QTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL3QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/a/e3f/wZwpmL1AzN1kDO3ATMwEDN0UTMyITNykTO0EDMwAjMwUzLwEzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
若,則稱為次同餘方程。
![同餘方程](/img/d/58f/wZwpmLyEjN4ITNyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同餘方程](/img/2/cf2/wZwpmL3gTN1IjMyMTOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzL4QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/d/58f/wZwpmLyEjN4ITNyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義2:設是整數,當時,成立,則稱是同餘方程的解。凡對於模同餘的解,被視為同一個解。同餘方程的解數是指它的關於模互不相余的所有解的個數,也即在模的一個完全剩餘系中的解的個數。
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
由定義2,同餘方程的解數不超過。
基本原理
定理:下面的結論成立:
![同餘方程](/img/3/07d/wZwpmLycTMxYTOyYDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL1QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1)設是整係數多項式,則同餘方程與
![同餘方程](/img/e/9ce/wZwpmLwczN0kTN4QjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0IzL3EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
等價;
![同餘方程](/img/9/b7b/wZwpmLxUTM4kDNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同餘方程](/img/4/4c9/wZwpmLzIzMxkDM3EDN3QTN1UTM1QDN5MjM5ADMwAjMwUzLxQzLyIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(2)設是整數,,則同餘方程與
![同餘方程](/img/d/a42/wZwpmL3cDM2IzMzkjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzLwczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
等價;
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/8/817/wZwpmLyQDO5UTN3IzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyMzLyUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同餘方程](/img/d/375/wZwpmL1AzNxEjM5IjM5kzM0UTMyITNykTO0EDMwAjMwUzLyIzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/8/b46/wZwpmLwIDMwYDNwIDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL0MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/d/58f/wZwpmLyEjN4ITNyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同餘方程](/img/6/d69/wZwpmLzMTN5cTM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/d/58f/wZwpmLyEjN4ITNyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(3)設是素數,,與都是整係數多項式,又設是同餘方程的解,則必是同餘方程
![同餘方程](/img/4/c17/wZwpmL1ATOwQzMyADN3QTN1UTM1QDN5MjM5ADMwAjMwUzLwQzL4AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
的解。
![同餘方程](/img/2/489/wZwpmLwYTMxIDN2cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL4czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/0/9c7/wZwpmLwETM1QTOwczM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3MzLyYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/0/9c7/wZwpmLwETM1QTOwczM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3MzLyYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/2/489/wZwpmLwYTMxIDN2cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL4czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
證明:(1)若,則成立,反之,若,則成立;
![同餘方程](/img/2/489/wZwpmLwYTMxIDN2cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL4czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/5/908/wZwpmLyYzN3ATN3MjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLzIzLygzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/5/908/wZwpmLyYzN3ATN3MjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLzIzLygzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同餘方程](/img/4/4c9/wZwpmLzIzMxkDM3EDN3QTN1UTM1QDN5MjM5ADMwAjMwUzLxQzLyIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/2/489/wZwpmLwYTMxIDN2cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL4czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2)若,則成立,反之,若,則由得成立;
![同餘方程](/img/5/11f/wZwpmLyEDOyYzM0UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLyUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同餘方程](/img/b/c3b/wZwpmLxczM5cjMwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同餘方程](/img/0/72f/wZwpmL4ITMzAzMyIjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyIzLyczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同餘方程](/img/5/7f2/wZwpmL4EDO1gzMyIjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyIzLzYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(3)若,則由是素數得或。證畢。