Algebraic Structures of Reduced Residue Systems 

In this section, we will introduce reduced residue systems, i.e. sets of integers which represent only those congruence classes modulo a positive integer $m > 0,$ which are co-prime to $m.$

We will also study their algebraic structure for prime and composite modules $m.$ Furthermore, we will prove theorems which characterize prime numbers, in particular, the Fermat’s little theorem, as well as the Wilson’s theorem.

| | | | created: 2019-05-12 09:15:43 | modified: 2019-06-22 08:36:28 | by: bookofproofs

1.Definition: Reduced Residue System 

2.Proposition: Creation of Reduced Residue Systems From Others 

3.Proposition: Existence and Number of Solutions of Congruence With One Variable 

4.Proposition: Multiplicative Group Modulo an Integer $(\mathbb Z_m^*,\cdot)$ 

5.Theorem: Euler-Fermat Theorem 

6.Proposition: A Necessary Condition for an Integer to be Prime 

7.Proposition: Wilson's Condition for an Integer to be Prime 

8.Proposition: Complete and Reduced Residue Systems (Revised)