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.

  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)

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