---
tag: 08B1RTW7VZ
kind: concept
name: Diophantine equation
msc: [11D99]
sources:
  - url: https://en.wikipedia.org/wiki/Diophantine_equation
    locator: lead section
same_as:
  wikidata: Q905896
  wikipedia: 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|01G1WY1GVX]] $x^2 + y^2 = z^2$, [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ and the equation of [[Fermat's Last Theorem|0Y5X3SDBCZ]].
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.