In this section, we will study *Diophantine equations*, i.e. equations involving rational numbers or integers, and methods to solve them.

| | | | created: 2014-02-20 21:51:49 | modified: 2019-06-23 15:48:33 | by: *bookofproofs*

## 1.**Definition**: Diophantine Equations

## 2.**Proposition**: Diophantine Equations of Congruences

## 3.**Proposition**: Existence of Solutions of an LDE With More Variables

## 4.**Proposition**: All Solutions Given a Solution of an LDE With Two Variables

## 5.**Theorem**: Chinese Remainder Theorem

## 6.**Proposition**: A Linear Term for 1 Using Two Co-prime Coefficients

## 7.**Proposition**: Counting the Solutions of Diophantine Equations of Congruences

## 8.**Proposition**: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number