Diophantine equation

Summary

A Diophantine equation is a polynomial equation with integer coefficients, for which one looks for integer solutions, or sometimes rational solutions. Typical questions ask whether a solution exists, whether there are finitely many, and how to find all of them.

Remarks

Examples are the linear equation $ax + by = c$, Pythagorean triples $x^2 + y^2 = z^2$, Pell's equation $x^2 - d y^2 = 1$ and the equation of Fermat's Last Theorem. By Hilbert's tenth problem, as solved by Yuri Matiyasevich in 1970, no algorithm decides for every Diophantine equation whether it has an integer solution. The name refers to Diophantus of Alexandria.

Explanations

L1 human

In a Diophantine equation, only whole numbers count as solutions. For example, $x^2 + y^2 = 3$ has real solutions, but no integer solutions. The restriction to integers makes such equations much harder than ordinary equations.