Diophantine Equation Generator

What this generator does

Draws two coefficients, then chooses the constant to make the equation either solvable or unsolvable as you asked. Where solutions exist it finds one with the extended Euclidean algorithm, lists the five nearest members of the family around it, and works out the smallest solution whose x is not negative.

How to use this tool

1. Choose how large the coefficients may be.
2. Decide whether you want a solvable equation or an impossible one.
3. Test solvability with the divisibility rule before searching.
4. Reveal the answer, and read the step between consecutive solutions off the table.

Understanding the controls

Largest number

How big the two coefficients may be, up to 200.

Want a solvable one

On, the constant is a multiple of the coefficients' greatest common divisor, so integer solutions exist. Off, it deliberately is not.

Seed

Reproduces the same equation exactly, so the solvable and unsolvable cases can be set side by side.

Show the answer

Reveals the base solution, a table of neighbouring solutions with the parameter t beside each, and the smallest one whose x is not negative — or states plainly that none exists.

Common use cases

• Practice at the solvability test before attempting a solution
• Showing that an equation can be perfectly ordinary and still have no integer answer
• Extended Euclidean algorithm exercises with verified answers
• Coin and measuring puzzles in their algebraic form
• Reproducing the same equation from a seed

How this generator works

The extended Euclidean algorithm produces x and y with ax + by = gcd(a, b), and multiplying both by the constant divided by that gcd solves the original. From there every solution is x + (b/g)t and y − (a/g)t, so the table is that formula evaluated at t from −2 to 2 — the step between consecutive rows is the content of the topic, and one pair on screen hides it. The smallest solution with x not negative is computed from the same formula rather than searched for. The check substitutes every listed solution back into the equation, because an evenly spaced table with the wrong step is still evenly spaced.

Randomness and fairness

The coefficients are random; solvability is decided by the divisibility rule, not by chance. Seeded equations reproduce exactly and are therefore explicitly not cryptographically secure. Unseeded, the browser's cryptographically secure generator supplies them.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

• Two unknowns and degree one only — the famous non-linear Diophantine problems are a different world entirely.
• Five solutions either side of the base one are listed, out of infinitely many; the rest follow from the same step.
• Coefficients are positive, so sign-handling cases do not appear.
• Solutions can be large negative numbers, which is correct but sometimes surprising.
• Equations are not stored between visits.

Privacy and your data

Coefficients and solutions are computed in your browser. Nothing about the equation or your seed is transmitted.