Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[ThueSolve] instead.

Let $f\left(x\&comma;y\right)$ be a rational binary form in $Z_{x,y}$, irreducible over $Q_{x,y}$, and m an integer. The Thue equation has the form $f\left(x\&comma;y\right)=m$ and the Thue inequality has the form $\left|f\left(x\&comma;y\right)\right|\le m$  (with $0<m$).

This function computes all solutions in $\mathbb{Z}\times \mathbb{Z}$ of the given Thue equation or inequality ex, subject to the constraint $\left|y\right|\le {10}^k$.  If k is omitted, it defaults to 10.

In the equation case, if there are no solutions with the above constraint and the last argument to thue is an unevaluated name (i.e., w is given), then to it will be assigned an expression sequence consisting of a diagnostic message and the integers m' in the range $-2\left|m\right|..2\left|m\right|$ such that the given Thue equation has a solution when m is replaced by m'.

In the inequality case, if the last argument to thue is an unevaluated name (i.e., w is given), then w is assigned the set of values obtained by substituting the various solutions to the inequality into the left hand side of the given Thue inequality.

Note that in the inequality case, ex must be given in the form $\left|f\right|\le m$ and not simply $f\le m$.

This function is part of the numtheory package, and so can be used in the form thue(..) only after performing the command with(numtheory). The function can always be accessed in the long form numtheory[thue](..).

$\mathrm{with}\left(\mathrm{numtheory}\right)\&colon;$

$\mathrm{thue}\left(x-5y=1\&comma;\left[x\&comma;y\right]\right)$

$\left[x=1+5\mathrm{\_Z1}\&comma;y=\mathrm{\_Z1}\right]$

$\mathrm{thue}\left(x^2+xy+y^2=19\&comma;\left[x\&comma;y\right]\right)$

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

$f≔x^3-3x{y}^2+y^3$

$f≔x^3-3x{y}^2+y^3$

$\mathrm{thue}\left(f=3\&comma;\left[x\&comma;y\right]\right)$

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

$\mathrm{thue}\left(f=2\&comma;\left[x\&comma;y\right]\&comma;'w'\right)$

$w$

$\mathrm{try\; the\; following\; constant\; term(s):},\mathrm{-3},\mathrm{-1},0,1,3$

$\mathrm{thue}\left(f=-3\&comma;\left[x\&comma;y\right]\&comma;20\&comma;'w'\right)$

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

$w$

$\mathrm{-3}$

$f≔\mathrm{abs}\left(x^3+x^{2}y-2x{y}^2-y^3\right)\&colon;$

$\mathrm{sols}≔\mathrm{thue}\left(f\le 5\&comma;\left[x\&comma;y\right]\&comma;'w'\right)$

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

$w$

$\mathrm{equality\; holds\; for\; the\; following\; constant\; term(s):},0,1$

$\mathrm{eval}\left(\mathrm{subs}\left(\mathrm{sols}\left[2\right]\&comma;f\right)\right)$

$1$

$g≔x^5+x^{4}y-4x^3{y}^2-3x^2{y}^3+3x{y}^4+y^5$

$g≔x^5+x^{4}y-4x^3{y}^2-3x^2{y}^3+3x{y}^4+y^5$

$\mathrm{thue}\left(\mathrm{abs}\left(g\right)\le 10\&comma;\left[x\&comma;y\right]\&comma;'w'\right)$

$\left[x=0\&comma;y=0\right],\left[x=\mathrm{-2}\&comma;y=1\right],\left[x=\mathrm{-1}\&comma;y=\mathrm{-1}\right],\left[x=\mathrm{-1}\&comma;y=0\right],\left[x=\mathrm{-1}\&comma;y=1\right],\left[x=0\&comma;y=\mathrm{-1}\right],\left[x=0\&comma;y=1\right],\left[x=1\&comma;y=\mathrm{-1}\right],\left[x=1\&comma;y=0\right],\left[x=1\&comma;y=1\right],\left[x=2\&comma;y=\mathrm{-1}\right]$

$w$

$\mathrm{equality\; holds\; for\; the\; following\; constant\; term(s):},0,1$

$\mathrm{thue}\left(\mathrm{abs}\left(x^3+y^3\right)\le 5\&comma;\left[x\&comma;y\right]\right)$

Error, (in numtheory:-thue) this binary form is not irreducible