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Diophantine Equations and Complete and Reduced Residue Systems

The following section is dedicated to further important concepts of elementary number theory:

  • Diophantine equations, i.e. equations involving integers only,
  • complete residue systems, i.e. sets of integers which represent all possible congruence classes modulo a positive integer $m > 0,$
  • and reduced residue systems, i.e. complete residue systems, from which those integers have been removed, which have common divisors with $m.$

The last two concepts will help us to get deeper insights for possible solutions of Diophantine equations. Knowing the solutions of such equations often enables us to solve practical applications which can be modeled by some of these equations. We start with some definitions and basic facts about Diophantine equations.

| | | | created: 2019-04-19 06:41:31 | modified: 2019-04-19 07:05:18 | by: bookofproofs | references: [1272], [8152]

1.Definition: Diophantine Equations

2.Proposition: Diophantine Equations of Congruences

3.Definition: Complete Residue System

4.Definition: Reduced Residue System

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

6.Proposition: Existence and Number of Solutions of an LDE With One Variable

7.Proposition: Existence of Solutions of an LDE With More Variables

8.Proposition: All Solutions Given a Solution of an LDE With Two Variables

9.Proposition: A Linear Term for 1 Using Two Co-prime Coefficients

10.Theorem: Chinese Remainder Theorem

11.Proposition: Counting the Solutions of Diophantine Equations of Congruences

12.Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number

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Bibliography (further reading)

[8152] Jones G., Jones M.: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] Landau, Edmund: “Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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