nLab Diophantine equation

Idea

A Diophantine equation is a polynomial equation over the integers, hence an equation of the form

$$P(x_1, x_2, \cdots , x_n) = 0$$

where $P(x_1, x_2, \cdots , x_n) \in \mathbb{Z}[x_1, \cdots, x_m]$ is a polynomial in $n$ variables with integer coefficients, and where solutions are considered for these variables taking values in the integers.

These are equations of central interest in number theory.

Spaces of solutions of Diophantine equations may be thought of as varieties in arithmetic geometry.

References

• Wikipedia, Diophantine equation