Thue Solve - Maple Help

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

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

The variables may be explicitly given.

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

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

 > $\mathrm{infolevel}\left[\mathrm{ThueSolve}\right]≔1$
 ${{\mathrm{infolevel}}}_{{\mathrm{ThueSolve}}}{≔}{1}$ (3)
 > $\mathrm{ThueSolve}\left({x}^{3}-3x{y}^{2}+{y}^{3}=2\right)$
 ThueSolve:   try the following constant terms   -3, -1, 0, 1, 3
 $\left[\right]$ (4)
 > $\mathrm{ThueSolve}\left(\mathrm{abs}\left({x}^{3}+{x}^{2}y-2x{y}^{2}-{y}^{3}\right)\le 5\right)$
 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.