Siegel identity

Short description: One of two formulae that are used in the resolution of Diophantine equations

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

[math]\displaystyle{ \frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 . }[/math]

The second is

[math]\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 . }[/math]

Application

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

See also

• Siegel formula

References

• Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 40. ISBN 0-521-20461-5. 
• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. 9. Cambridge University Press. p. 53. ISBN 978-0-521-88268-2. 
• Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. 244. ISBN 0-387-90517-0. 
• Lang, Serge (1978). Elliptic Curves: Diophantine Analysis. Grundlehren der mathematischen Wissenschaften. 231. Springer-Verlag. ISBN 0-387-08489-4. 
• Smart, N. P. (1998). The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts. 41. Cambridge University Press. pp. 36–37. ISBN 0-521-64633-2. https://archive.org/details/algorithmicresol0000smar/page/36.