# Thue equation

In mathematics, a **Thue equation** is a Diophantine equation of the form

*ƒ*(*x*,*y*) =*r*,

where *ƒ* is an irreducible bivariate 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*.^{[1]}

The Thue equation is solvable effectively: there is an explicit bound on the solutions *x*, *y* of the form where constants *C*_{1} and *C*_{2} 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.^{[2]}

## Solving Thue equations[edit]

Solving a Thue equation can be described as an algorithm^{[3]} ready for implementation in software. In particular, it is implemented in the following computer algebra systems:

- in PARI/GP as functions
*thueinit()*and*thue()*. - in Magma computer algebra system as functions
*ThueObject()*and*ThueSolve()*. - in Mathematica through
*Reduce*

## See also[edit]

## References[edit]

**^**A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen".*Journal für die reine und angewandte Mathematik*.**1909**(135): 284–305. doi:10.1515/crll.1909.135.284.**^**Baker, Alan (1975).*Transcendental Number Theory*. Cambridge University Press. p. 38. ISBN 0-521-20461-5.**^**N. Tzanakis and B. M. M. de Weger (1989). "On the practical solution of the Thue equation".*Journal of Number Theory*.**31**(2): 99–132. doi:10.1016/0022-314X(89)90014-0.

## Further reading[edit]

- Baker, Alan; Wüstholz, Gisbert (2007).
*Logarithmic Forms and Diophantine Geometry*. New Mathematical Monographs.**9**. Cambridge University Press. ISBN 978-0-521-88268-2.

This number theory-related article is a stub. You can help Wikipedia by expanding it. |