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 soluble effectively: there is an explicit bound on the solutions x, y of the form $(C_{1}r)^{{C_{2}}}$ where constants C₁ and C₂ 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

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

References

1. A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen". Journal für die reine und angewandte Mathematik. 135: 284–305. doi:10.1515/crll.1909.135.284.
2. Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 38. ISBN 0-521-20461-5.
3. 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

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