diophantine equations calculator

What this diophantine equations calculator does

A Diophantine equation asks for integer solutions rather than decimal or real-number answers. This tool focuses on the most common case: the linear equation in two variables, ax + by = c.

Instead of giving just one answer, the calculator tells you whether:

• no integer solution exists,
• exactly constrained forms exist (through a parameter), or
• every integer pair works (in the special 0x + 0y = 0 case).

How to use the calculator

Step 1: Enter coefficients

Type integer values for a, b, and c. These can be positive, negative, or zero.

Step 2: Set a sample parameter range

If solutions exist, the calculator expresses them using parameter t. Set a range (for example, -3 to 3) to view concrete sample pairs.

Step 3: Click “Solve Equation”

You’ll get:

• the gcd condition check,
• a particular integer solution,
• the general solution formula,
• and a table of sample integer solutions.

The math behind the result

The gcd test

Let g = gcd(a, b). The equation ax + by = c has an integer solution if and only if g divides c. This is the most important test in linear number theory for two variables.

Extended Euclidean Algorithm

When g | c, we use the Extended Euclidean Algorithm to find integers u and v such that: au + bv = g. Multiplying by c/g gives one particular solution (x₀, y₀).

General integer solution

Once one solution is known, all integer solutions are:

x = x₀ + (b/g)t
y = y₀ - (a/g)t
where t ∈ Z.

Example walkthrough

Try 15x + 21y = 84. Here, gcd(15, 21) = 3, and 3 divides 84, so solutions exist. The calculator finds one valid pair and then generates all others with the parameter formula.

If you change it to 15x + 21y = 85, then 3 does not divide 85, so no integer solution exists.

Common use cases

• Checking solvability constraints in number theory homework.
• Building integer combinations in coding and algorithm challenges.
• Working with modular arithmetic and congruence transformations.
• Testing integer feasibility before optimization steps.

Limitations

This calculator solves linear two-variable Diophantine equations only. It does not directly solve nonlinear forms like x² + y² = n or systems with many variables. For those cases, additional theory or computational search methods are required.

Quick FAQ

Why do I get infinitely many answers?

Because once one integer solution exists for ax + by = c, moving by the step vector (b/g, -a/g) preserves equality, producing new integer solutions forever.

What if one coefficient is zero?

The calculator handles it automatically. For example, if a = 0, the equation becomes by = c, so solvability depends on whether b divides c.

Can I use large integers?

Yes. This page uses integer-safe BigInt arithmetic for core calculations.