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

• Definition:Diophantine Problem

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

• Results about Diophantine equations can be found here.

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. $\text {370?}$ – $\text {415}$)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Diophantine equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Diophantine equation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Diophantine equation