Diophantine Equation Solver
ax + by = c
Solution: -
General Form: -
Understanding Diophantine Equations
What are Diophantine Equations?
Diophantine equations are polynomial equations that require integer solutions.
Key Properties
- Linear Form: ax + by = c
- Bézout's Identity: gcd(a,b) = ax + by
- Solution Existence: c must be divisible by gcd(a,b)
- General Solution: x = x₀ + (b/d)t, y = y₀ - (a/d)t
- where d = gcd(a,b) and t is an integer
Types and Methods
Linear Equations
Extended Euclidean algorithm
Modular arithmetic
Solution parametrization
Integer lattices
Quadratic Forms
Pell's equation
Circle method
Reduction theory
Class numbers
Higher Degree
Thue equations
Elliptic curves
Fermat's Last Theorem
ABC conjecture
Applications
Cryptography
Combinatorics
Computer science
Number theory
Advanced Topics
Algebraic Theory
Ideal theory
Ring theory
Algebraic numbers
Galois theory
Analytic Methods
L-functions
Zeta functions
Height functions
Density theorems
Geometric Aspects
Lattice points
Minkowski theory
Algebraic geometry
Arithmetic surfaces
Computational Methods
LLL algorithm
Continued fractions
Reduction algorithms
Sieve methods