# Siegel identity

In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.

## Statement

The first formula is

${\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}=1.}$

The second is

${\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{2}}{t-x_{3}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{1}}{t-x_{3}}}=1.}$

## Application

The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.