solve equations for integer solutions - Maple Help

isolve - solve equations for integer solutions

 Calling Sequence isolve(eqns, vars)

Parameters

 eqns - set of equations or single equation, or inequalities vars - (optional) set of variables or a variable

Description

 • The procedure isolve solves the equations in eqns over the integers. It solves for all of the indeterminates occurring in the equations.
 • The optional second argument vars is used to name global variables that have integer values and occur in the solution, and if there is only one argument, then the global names _Z1, _Z2, and so forth, are used. For non-negative solutions, _NN1, _NN2, and so forth, are used. This has been introduced since Release 5.1 which used to use _N1 in order to make the behavior similar to that of solve.
 • It returns the NULL value if either there are no integer solutions or Maple is unable to find the solutions.
 • The isolve command has some limited ability to deal with inequalities.

Examples

 > $\mathrm{isolve}\left(3x-4y=7\right)$
 $\left\{{x}{=}{5}{+}{4}{}{\mathrm{_Z1}}{,}{y}{=}{2}{+}{3}{}{\mathrm{_Z1}}\right\}$ (1)
 > $\mathrm{isolve}\left(3x-4y=7,a\right)$
 $\left\{{x}{=}{5}{+}{4}{}{a}{,}{y}{=}{2}{+}{3}{}{a}\right\}$ (2)
 > $\mathrm{isolve}\left(\left\{3x-4y=7,x+y=14\right\}\right)$
 $\left\{{x}{=}{9}{,}{y}{=}{5}\right\}$ (3)

NULL is returned if Maple is unable to find any integer solutions.

 > $\mathrm{isolve}\left({x}^{2}=3\right)$
 > $\mathrm{solve}\left(\left\{4x-y=7,x+2y=8\right\}\right)$
 $\left\{{x}{=}\frac{{22}}{{9}}{,}{y}{=}\frac{{25}}{{9}}\right\}$ (4)
 > $\mathrm{isolve}\left(\left\{4x-y=7,x+2y=8\right\}\right)$

The following homogeneous polynomial in x, y, z has genus 0.

 > $\mathrm{isolve}\left({y}^{4}-{z}^{2}{y}^{2}-3xz{y}^{2}-{x}^{3}z\right)$
 $\left\{{x}{=}\frac{{\mathrm{_Z3}}{}{{\mathrm{_Z1}}}^{{2}}{}\left({{\mathrm{_Z1}}}^{{2}}{-}{{\mathrm{_Z2}}}^{{2}}\right)}{{\mathrm{igcd}}{}\left({{\mathrm{_Z1}}}^{{2}}{}\left({{\mathrm{_Z1}}}^{{2}}{-}{{\mathrm{_Z2}}}^{{2}}\right){,}{-}{{\mathrm{_Z1}}}^{{3}}{}{\mathrm{_Z2}}{,}{{\mathrm{_Z2}}}^{{4}}\right)}{,}{y}{=}{-}\frac{{\mathrm{_Z3}}{}{{\mathrm{_Z1}}}^{{3}}{}{\mathrm{_Z2}}}{{\mathrm{igcd}}{}\left({{\mathrm{_Z1}}}^{{2}}{}\left({{\mathrm{_Z1}}}^{{2}}{-}{{\mathrm{_Z2}}}^{{2}}\right){,}{-}{{\mathrm{_Z1}}}^{{3}}{}{\mathrm{_Z2}}{,}{{\mathrm{_Z2}}}^{{4}}\right)}{,}{z}{=}\frac{{\mathrm{_Z3}}{}{{\mathrm{_Z2}}}^{{4}}}{{\mathrm{igcd}}{}\left({{\mathrm{_Z1}}}^{{2}}{}\left({{\mathrm{_Z1}}}^{{2}}{-}{{\mathrm{_Z2}}}^{{2}}\right){,}{-}{{\mathrm{_Z1}}}^{{3}}{}{\mathrm{_Z2}}{,}{{\mathrm{_Z2}}}^{{4}}\right)}\right\}$ (5)
 > $\mathrm{isolve}\left({y}^{4}-{z}^{2}{y}^{2}-3xz{y}^{2}-{x}^{3}z,\left\{a,b,c\right\}\right)$
 $\left\{{x}{=}\frac{{c}{}{{a}}^{{2}}{}\left({{a}}^{{2}}{-}{{b}}^{{2}}\right)}{{\mathrm{igcd}}{}\left({{a}}^{{2}}{}\left({{a}}^{{2}}{-}{{b}}^{{2}}\right){,}{-}{{a}}^{{3}}{}{b}{,}{{b}}^{{4}}\right)}{,}{y}{=}{-}\frac{{c}{}{{a}}^{{3}}{}{b}}{{\mathrm{igcd}}{}\left({{a}}^{{2}}{}\left({{a}}^{{2}}{-}{{b}}^{{2}}\right){,}{-}{{a}}^{{3}}{}{b}{,}{{b}}^{{4}}\right)}{,}{z}{=}\frac{{c}{}{{b}}^{{4}}}{{\mathrm{igcd}}{}\left({{a}}^{{2}}{}\left({{a}}^{{2}}{-}{{b}}^{{2}}\right){,}{-}{{a}}^{{3}}{}{b}{,}{{b}}^{{4}}\right)}\right\}$ (6)
 > $\left\{\mathrm{isolve}\right\}\left(\left\{c=d,d=1,b\le 1,-b\le 0,a<1,-a\le 0\right\}\right)$
 $\left\{\left\{{a}{=}{0}{,}{b}{=}{0}{,}{c}{=}{1}{,}{d}{=}{1}\right\}{,}\left\{{a}{=}{0}{,}{b}{=}{1}{,}{c}{=}{1}{,}{d}{=}{1}\right\}\right\}$ (7)