The AreCoprime function tests whether a sequence of numbers is relatively prime in a given domain. A sequence of numbers are relatively prime (or coprime) if the greatest common divisor of the numbers is equal to 1.

By default, the test is performed in the integer domain (that is, domain = integer). To test whether a sequence of Gaussian integers is relatively prime, use either domain = GaussInt or domain = gaussian for domain_opt.

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

$\mathrm{AreCoprime}\left(4\,9\right)$

$\mathrm{true}$

$\mathrm{igcd}\left(4\,9\right)$

$1$

$\mathrm{AreCoprime}\left(14\,21\right)$

$\mathrm{false}$

$\mathrm{igcd}\left(14\,21\right)$

$7$

$\mathrm{AreCoprime}\left(1+2\mathrm{I}\,1-2\mathrm{I}\,\mathrm{domain}=\mathrm{gaussian}\right)$

$\mathrm{true}$

$\mathrm{GaussInt}:-\mathrm{GIgcd}\left(1+2\mathrm{I}\,1-2\mathrm{I}\right)$

$1$

$\mathrm{AreCoprime}\left(-3+5\mathrm{I}\,4+8\mathrm{I}\,\mathrm{domain}=\mathrm{gaussian}\right)$

$\mathrm{false}$

$\mathrm{GaussInt}:-\mathrm{GIgcd}\left(-3+5\mathrm{I}\,4+8\mathrm{I}\right)$

$1+\mathrm{I}$

$\mathrm{mat}≔\mathrm{Matrix}\left(15\,\left(i\,j\right)\mapsto \mathrm{`if`}\left(\mathrm{AreCoprime}\left(i\,j\right)\,1\,0\right)\right)$

$\mathrm{mat}≔\begin{array}{c}\left[\begin{array}{ccccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1& \dots \\ 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& \dots \\ 1& 1& 0& 1& 1& 0& 1& 1& 0& 1& \dots \\ 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& \dots \\ 1& 1& 1& 1& 0& 1& 1& 1& 1& 0& \dots \\ 1& 0& 0& 0& 1& 0& 1& 0& 0& 0& \dots \\ 1& 1& 1& 1& 1& 1& 0& 1& 1& 1& \dots \\ 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& \dots \\ 1& 1& 0& 1& 1& 0& 1& 1& 0& 1& \dots \\ 1& 0& 1& 0& 0& 0& 1& 0& 1& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{15 × 15 Matrix}\end{array}$

$\mathrm{Statistics}:-\mathrm{HeatMap}\left(\mathrm{mat}\,\mathrm{color}=\left[''White''\,''DarkRed''\right]\right)$

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

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