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NumberTheory

 ThueSolve
 solutions to a Thue equation or inequality

 Calling Sequence ThueSolve(expr) ThueSolve(expr, bound = b) ThueSolve(expr, vars, bound = b)

Parameters

 expr - Thue equation or Thue inequality vars - set of two names bound = b - (optional) keyword argument where b is a positive integer; defaults to $10$

Description

 • The ThueSolve function computes all the solutions to a Thue equation or inequality.
 • Let $f\left(x,y\right)$ be a binary form with integer coefficients and irreducible over the rationals. A binary form is a bivariate polynomial where every term has the same degree. Let $m$ be an integer. A Thue equation has the form $f\left(x,y\right)=m$ and a Thue inequality has the form $\left|f\left(x,y\right)\right|\le m$.
 • If the degree of $f$ is less than $3$, then isolve is used to find the solutions. Otherwise, there exists a finite number of solutions and this command finds all solutions $\left(x,y\right)$ given the constraint $\left|y\right|\le {10}^{\mathrm{bound}}$.

Examples

 > with(NumberTheory):
 > ThueSolve(x^2 + x*y + y^2 = 19);
 $\left[\left[{x}{=}{-5}{,}{y}{=}{2}\right]{,}\left[{x}{=}{-5}{,}{y}{=}{3}\right]{,}\left[{x}{=}{-3}{,}{y}{=}{-2}\right]{,}\left[{x}{=}{-3}{,}{y}{=}{5}\right]{,}\left[{x}{=}{-2}{,}{y}{=}{-3}\right]{,}\left[{x}{=}{-2}{,}{y}{=}{5}\right]{,}\left[{x}{=}{2}{,}{y}{=}{-5}\right]{,}\left[{x}{=}{2}{,}{y}{=}{3}\right]{,}\left[{x}{=}{3}{,}{y}{=}{-5}\right]{,}\left[{x}{=}{3}{,}{y}{=}{2}\right]{,}\left[{x}{=}{5}{,}{y}{=}{-3}\right]{,}\left[{x}{=}{5}{,}{y}{=}{-2}\right]\right]$ (1)

The variables may be explicitly given.

 > ThueSolve(x^3-3*x*y^2+y^3 = 3, {x, y});
 $\left[\left[{x}{=}{-1}{,}{y}{=}{-2}\right]{,}\left[{x}{=}{-1}{,}{y}{=}{1}\right]{,}\left[{x}{=}{2}{,}{y}{=}{1}\right]\right]$ (2)

Setting infolevel to $1$ or greater will give additional information when solutions do not exist or when solving a Thue inequality.

 > infolevel[ThueSolve] := 1;
 ${{\mathrm{infolevel}}}_{{\mathrm{ThueSolve}}}{≔}{1}$ (3)
 > ThueSolve(x^3-3*x*y^2+y^3 = 2);
 ThueSolve:   try the following constant terms   -3, -1, 0, 1, 3
 $\left[\right]$ (4)
 > ThueSolve(abs(x^3+x^2*y-2*x*y^2-y^3) <= 5);
 ThueSolve:   equality holds for the follow constant terms   0, 1
 $\left[\left[{x}{=}{0}{,}{y}{=}{0}\right]{,}\left[{x}{=}{-9}{,}{y}{=}{5}\right]{,}\left[{x}{=}{-5}{,}{y}{=}{-4}\right]{,}\left[{x}{=}{-4}{,}{y}{=}{9}\right]{,}\left[{x}{=}{-2}{,}{y}{=}{1}\right]{,}\left[{x}{=}{-1}{,}{y}{=}{-1}\right]{,}\left[{x}{=}{-1}{,}{y}{=}{0}\right]{,}\left[{x}{=}{-1}{,}{y}{=}{1}\right]{,}\left[{x}{=}{-1}{,}{y}{=}{2}\right]{,}\left[{x}{=}{0}{,}{y}{=}{-1}\right]{,}\left[{x}{=}{0}{,}{y}{=}{1}\right]{,}\left[{x}{=}{1}{,}{y}{=}{-2}\right]{,}\left[{x}{=}{1}{,}{y}{=}{-1}\right]{,}\left[{x}{=}{1}{,}{y}{=}{0}\right]{,}\left[{x}{=}{1}{,}{y}{=}{1}\right]{,}\left[{x}{=}{2}{,}{y}{=}{-1}\right]{,}\left[{x}{=}{4}{,}{y}{=}{-9}\right]{,}\left[{x}{=}{5}{,}{y}{=}{4}\right]{,}\left[{x}{=}{9}{,}{y}{=}{-5}\right]\right]$ (5)

Compatibility

 • The NumberTheory[ThueSolve] command was introduced in Maple 2016.
 • For more information on Maple 2016 changes, see Updates in Maple 2016.