galois.totatives(n: int) → list[int]

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

Parameters:
n: int

A positive integer.

Returns:

The totatives of \(n\).

See also

euler_phi, carmichael_lambda, is_cyclic

Notes

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

References

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