Diophantine Equations

Diophantine equations are polynomial equations whose solutions are required to be integers. Named after the ancient mathematician Diophantus, they appear in number theory, cryptography, and many combinatorial problems.

Common Types

Linear Diophantine equations: ax + by = c
Pell’s equation: x² − Dy² = 1
Fermat’s Last Theorem: xⁿ + yⁿ = zⁿ for n > 2 (no non‑trivial integer solutions)

Linear Example

Find integer solutions to 15x + 21y = 3.

gcd(15,21)=3 → solution exists.
One solution: x=−1, y=1.
General solution: x=−1+7t, y=1−5t (t∈ℤ)

Interactive Linear Solver

Enter coefficients a, b and constant c to find a particular integer solution (if one exists).

a:
b:
c: