In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Siegel's identity refers to one of two formulae that are used in the resolution of
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s.
Statement
The first formula is
:
The second is
:
Application
The identities are used in translating Diophantine problems connected with integral points on
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
s into
S-unit equation
In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units.
Definition
Let ''K'' be a number ...
s.
See also
*
Siegel formula
References
*
*
*
*
* {{cite book , title=The Algorithmic Resolution of Diophantine Equations , volume=41 , series=London Mathematical Society Student Texts , first=N. P. , last=Smart , authorlink=Nigel Smart (cryptographer) , publisher=
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, year=1998 , isbn=0-521-64633-2 , page
36–37, url=https://archive.org/details/algorithmicresol0000smar/page/36
Algebraic identities
Diophantine equations