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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

A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen". Journal für die reine und angewandte Mathematik 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.