Returns the positive integers (totatives) in \([1, n)\) that are coprime to \(n\).

The totatives of \(n\) form the multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\).

Parameters¶

n: int¶

A positive integer.

Returns¶

The totatives of \(n\).

References¶

• Section 2.4.3 from https://cacr.uwaterloo.ca/hac/about/chap2.pdf

• https://oeis.org/A000010

Examples¶

Find the totatives that are coprime with \(n = 20\).

In [1]: n = 20

In [2]: x = galois.totatives(n); x
Out[2]: [1, 3, 7, 9, 11, 13, 17, 19]

The number of totatives of \(n\) is \(\phi(n)\).

In [3]: phi = galois.euler_phi(n); phi
Out[3]: 8

In [4]: len(x) == phi
Out[4]: True