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{{For|other theorems named after Axel Thue|Thue's theorem (disambiguation)}}

In [[mathematics]], a '''Thue equation''' is a [[Diophantine equation]] of the form 

:''&fnof;''(''x'',''y'') = ''r'', 

where ''ƒ'' is an [[irreducible polynomial|irreducible]] bivariate [[Algebraic form|form]] of degree at least 3 over the rational numbers, and ''r'' is a nonzero [[rational number]]. It is named after [[Axel Thue]] who in 1909 proved a [[theorem]], now called '''Thue's theorem''', that a Thue equation has finitely many solutions in integers ''x'' and ''y''.<ref>{{cite journal |author=A. Thue |title=Über Annäherungswerte algebraischer Zahlen |journal=[[Journal für die reine und angewandte Mathematik]] |volume=135 |pages=284–305 |year=1909 |url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0135}}</ref>

The Thue equation is [[Effective results in number theory|soluble effectively]]: there is an explicit bound on the solutions ''x'', ''y'' of the form <math>(C_1 r)^{C_2}</math> where constants ''C''<sub>1</sub> and ''C''<sub>2</sub> depend only on the form ''ƒ''.   A stronger result holds, that if ''K'' is the field generated by the roots of ''ƒ'' then the equation has only finitely many solutions with ''x'' and ''y'' integers of ''K'' and again these may be effectively determined.<ref>{{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | page=38 }}</ref>

==Solving Thue equations==
Solving a Thue equation can be described as an algorithm<ref>{{cite journal |author=N. Tzanakis and B. M. M. de Weger |title=On the practical solution of the Thue equation |journal=[[Journal of Number Theory]] |volume=31 |issue=2 |year=1989 |pages=99–132 |doi=10.1016/0022-314X(89)90014-0}}</ref> ready for implementation in software. In particular, it is implemented in the following [[computer algebra system]]s:
* in [[PARI/GP]] as functions ''thueinit()'' and ''thue()''.
* in [[Magma computer algebra system]] as functions ''ThueObject()'' and ''ThueSolve()''.
* in [[Mathematica]] through ''Reduce'' 

==References==
{{reflist}}

==Further reading==
* {{cite book | first1=Alan | last1=Baker | authorlink1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | authorlink2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 }}

[[Category:Diophantine equations]]
[[Category:Theorems in number theory]]

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