◀ ▲ ▶Branches / Number-theory / Section: Algebraic Structures of Reduced Residue Systems

Section: 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.

Table of Contents

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)

Thank you to the contributors under CC BY-SA 4.0!

Github: