Diophantine Equation Calculator
Find integer solutions to Ax + By = C with step-by-step insights and verification notes from number theory.
Equation Inputs
Provide integer coefficients for A, B, and the constant C to see a representative solution and the general parameterized family.
Integer solutions appear when gcd(A, B) divides C.
How to Use This Calculator
This calculator finds integer solutions to linear Diophantine equations of the form Ax + By = C. Enter integer values for A, B, and C, then click Calculate to view one particular solution along with the generalized formula for every integer pair that satisfies the equation.
Methodology
The engine relies on the Extended Euclidean Algorithm to compute a base solution where A times x plus B times y equals the greatest common divisor of A and B. The base solution is then scaled by C divided by that gcd, which yields one valid integer solution. Because any solution can be shifted by multiples of (B/d, -A/d), the tool also displays the general solution family.
Source and interpretation
All calculations are based on standard number theory techniques. For deeper background on linear Diophantine equations and the extended Euclidean approach, this source is a helpful guide.
Glossary of terms
- Coefficient A and B: The integers multiplying variables x and y.
- Constant C: The value on the right-hand side of the equation that the combination must equal.
- Solutions: Integer values for x and y that satisfy the equality.
Example
For 3x + 4y = 5, the calculator returns one solution (such as x = 3, y = -1) plus the general form x = 3 + 4t, y = -1 - 3t that generates every integer pair satisfying the equation.
Frequently Asked Questions
What are Diophantine equations?
Polynomial equations for which only integer solutions are of interest.
Why are they important?
Number theorists, cryptographers, and algorithm designers use Diophantine techniques to reason about integer constraints.
Can all Diophantine equations be solved?
No; some forms are undecidable or require specialized methods. Linear Diophantine equations like Ax + By = C always have solutions when gcd(A, B) divides C.
What is an example of a Diophantine equation?
Fermat's Last Theorem is a famous historical example of a Diophantine problem with deep complexity.
Where can I learn more?
Explore academic notes, textbooks, and educational websites for number theory, especially resources that cover the Euclidean and Extended Euclidean algorithms.