Definition:Diophantine Equation

A Diophantine Equation is an indeterminate polynomial equation that allows the variables to take integer values only. They define an algebraic curve, algebraic surface, or more general object and ask about the lattice points on it.

The term Diophantine refers to Diophantus of Alexandria.

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:

$$ax + by + c = 0$$

$$a_1 x_1 + a^2 x_2 + \cdots + a_n x_n = b$$

The Fermat Connection
$$x^n + y^n = z^n$$ is a common example of a Diophantine Equation, for which Fermat's Last Theorem was a conjecture regarding its nature of existence.