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 $\left|y\right|\le {10}^{\mathrm{bound}}$.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{ThueSolve}\left(x^2+xy+y^2=19\right)$

$\left[\left[x=\mathrm{-5}\,y=2\right]\,\left[x=\mathrm{-5}\,y=3\right]\,\left[x=\mathrm{-3}\,y=\mathrm{-2}\right]\,\left[x=\mathrm{-3}\,y=5\right]\,\left[x=\mathrm{-2}\,y=\mathrm{-3}\right]\,\left[x=\mathrm{-2}\,y=5\right]\,\left[x=2\,y=\mathrm{-5}\right]\,\left[x=2\,y=3\right]\,\left[x=3\,y=\mathrm{-5}\right]\,\left[x=3\,y=2\right]\,\left[x=5\,y=\mathrm{-3}\right]\,\left[x=5\,y=\mathrm{-2}\right]\right]$

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

$\left[\left[x=\mathrm{-1}\,y=\mathrm{-2}\right]\,\left[x=\mathrm{-1}\,y=1\right]\,\left[x=2\,y=1\right]\right]$

$\mathrm{infolevel}\left[\mathrm{ThueSolve}\right]≔1$

${\mathrm{infolevel}}_{\mathrm{ThueSolve}}≔1$

$\mathrm{ThueSolve}\left(x^3-3x{y}^2+y^3=2\right)$

ThueSolve:   try the following constant terms   -3, -1, 0, 1, 3

$\mathrm{ThueSolve}\left(\mathrm{abs}\left(x^3+x^{2}y-2x{y}^2-y^3\right)\le 5\right)$

ThueSolve:   equality holds for the follow constant terms   0, 1

$\left[\left[x=0\,y=0\right]\,\left[x=\mathrm{-9}\,y=5\right]\,\left[x=\mathrm{-5}\,y=\mathrm{-4}\right]\,\left[x=\mathrm{-4}\,y=9\right]\,\left[x=\mathrm{-2}\,y=1\right]\,\left[x=\mathrm{-1}\,y=\mathrm{-1}\right]\,\left[x=\mathrm{-1}\,y=0\right]\,\left[x=\mathrm{-1}\,y=1\right]\,\left[x=\mathrm{-1}\,y=2\right]\,\left[x=0\,y=\mathrm{-1}\right]\,\left[x=0\,y=1\right]\,\left[x=1\,y=\mathrm{-2}\right]\,\left[x=1\,y=\mathrm{-1}\right]\,\left[x=1\,y=0\right]\,\left[x=1\,y=1\right]\,\left[x=2\,y=\mathrm{-1}\right]\,\left[x=4\,y=\mathrm{-9}\right]\,\left[x=5\,y=4\right]\,\left[x=9\,y=\mathrm{-5}\right]\right]$

The NumberTheory[ThueSolve] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.