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NumberTheory

 RootsOfUnity
 modular roots of unity

 Calling Sequence RootsOfUnity(k, n)

Parameters

 k - prime number n - positive integer

Description

 • The RootsOfUnity(k, n) command computes all the kth roots of unity modulo n.
 • An integer $x$ is said to be a $k$th root of unity modulo $n$ if ${x}^{k}=1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n$.

Examples

 > with(NumberTheory):
 > unity := RootsOfUnity(5, 8965);
 ${\mathrm{unity}}{≔}\left\{{1}{,}{1631}{,}{2446}{,}{3261}{,}{6521}\right\}$ (1)
 > map(x -> x^5 mod 8965, unity);
 $\left\{{1}\right\}$ (2)

Distribution of the second roots of unity. A point $x,y$ on the plot denotes that $y$ is a second root of unity modulo $x$.

 > plots:-pointplot(select(p -> p^2 mod p = 1, [seq(seq([i,j], j=0..(i-1)), i = 1..1000)]), labels=["Modulus", "Second roots of unity"], labeldirections=[horizontal, vertical]); Compatibility

 • The NumberTheory[RootsOfUnity] command was introduced in Maple 2016.