Thue equation

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

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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 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:

See also

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References

  1. 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.
  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