Diophantine Solver
Integer solutions of ax+by=c
About this calculator
The Diophantine Solver finds all integer solutions of the linear equation a·x + b·y = c. It uses the extended Euclidean algorithm to compute gcd(a, b) and Bézout coefficients, reports whether a solution exists, and gives the full one-parameter family x = x0 + (b/g)·t, y = y0 − (a/g)·t with sample solutions and a lattice-point visual.
How to use the Diophantine Solver calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- 3x + 5y = 11 → x = 2 + 5t, y = 1 − 3t
- 6x + 9y = 15 → gcd 3 divides 15, x = 4 + 3t, y = −1 − 2t
- 2x + 4y = 5 → gcd 2 does not divide 5, no integer solutions
- 0x + 5y = 10 → y = 2, x is free (x = t)
- 0x + 0y = 0 → every integer pair (x, y) is a solution
Frequently asked questions
How does the Diophantine Solver calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Diophantine Solver calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.