# Definition:Diophantine Equation

## Definition

A **Diophantine equation** is an indeterminate polynomial equation that allows the variables to take integer values only.

It defines an algebraic curve, algebraic surface, or more general object, and asks about the lattice points on it.

### Linear Diophantine Equation

A **linear Diophantine equation** is a Diophantine equation in which all the arguments appear to no higher than the first degree.

For example:

- $ax + by + c = 0$

- $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$

## Also see

- Fermat's Last Theorem: $x^n + y^n = z^n$ is a well-known example of a
**Diophantine equation**.

## Source of Name

This entry was named for Diophantus of Alexandria.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.9$: Hypatia (A.D. $370?$ – $415$)