galois.primitive_roots

galois.primitive_roots(n, start=1, stop=None, reverse=False)

Finds all primitive roots modulo \(n\).

Parameters
n : int

A positive integer.

start : int

Starting value (inclusive) in the search for a primitive root. The default is 1. The resulting primitive roots, if found, will be \(\textrm{start} \le x < \textrm{stop}\).

stop : Optional[int]

Stopping value (exclusive) in the search for a primitive root. The default is None which corresponds to n. The resulting primitive roots, if found, will be \(\textrm{start} \le x < \textrm{stop}\).

reverse : bool

List all primitive roots in descending order, i.e. largest to smallest. Default is False.

Returns

All the positive primitive \(n\)-th roots of unity, \(x\).

Return type

List[int]

Notes

The integer \(g\) is a primitive root modulo \(n\) if the totatives of \(n\), the positive integers \(1 \le a < n\) that are coprime with \(n\), can be generated by powers of \(g\).

Alternatively said, \(g\) is a primitive root modulo \(n\) if and only if \(g\) is a generator of the multiplicative group of integers modulo \(n\),

\[(\mathbb{Z}/n\mathbb{Z}){^\times} = \{1, g, g^2, \dots, g^{\phi(n)-1}\}\]

where \(\phi(n)\) is order of the group.

If \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic, the number of primitive roots modulo \(n\) is given by \(\phi(\phi(n))\).

References

Examples

The elements of \((\mathbb{Z}/n\mathbb{Z}){^\times}\) are the positive integers less than \(n\) that are coprime with \(n\). For example, \((\mathbb{Z}/14\mathbb{Z}){^\times} = \{1, 3, 5, 9, 11, 13\}\).

# n is of type 2*p^k, which is cyclic
In [1]: n = 14

In [2]: galois.is_cyclic(n)
Out[2]: True

# The congruence class coprime with n
In [3]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[3]: {1, 3, 5, 9, 11, 13}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [4]: phi = galois.euler_phi(n); phi
Out[4]: 6

In [5]: len(Znx) == phi
Out[5]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [6]: for a in Znx:
   ...:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   ...:     primitive_root = span == Znx
   ...:     print("Element: {:2d}, Span: {:<20}, Primitive root: {}".format(a, str(span), primitive_root))
   ...: 
Element:  1, Span: {1}                 , Primitive root: False
Element:  3, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  5, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  9, Span: {9, 11, 1}          , Primitive root: False
Element: 11, Span: {9, 11, 1}          , Primitive root: False
Element: 13, Span: {1, 13}             , Primitive root: False

# Find the smallest primitive root
In [7]: galois.primitive_root(n)
Out[7]: 3

# Find all primitive roots
In [8]: roots = galois.primitive_roots(n); roots
Out[8]: [3, 5]

# Euler's totient function ϕ(ϕ(n)) counts the primitive roots of n
In [9]: len(roots) == galois.euler_phi(phi)
Out[9]: True

A counterexample is \(n = 15 = 3 \cdot 5\), which doesn’t fit the condition for cyclicness. \((\mathbb{Z}/15\mathbb{Z}){^\times} = \{1, 2, 4, 7, 8, 11, 13, 14\}\).

# n is of type p1^k1 * p2^k2, which is not cyclic
In [10]: n = 15

In [11]: galois.is_cyclic(n)
Out[11]: False

# The congruence class coprime with n
In [12]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[12]: {1, 2, 4, 7, 8, 11, 13, 14}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [13]: phi = galois.euler_phi(n); phi
Out[13]: 8

In [14]: len(Znx) == phi
Out[14]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [15]: for a in Znx:
   ....:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   ....:     primitive_root = span == Znx
   ....:     print("Element: {:2d}, Span: {:<13}, Primitive root: {}".format(a, str(span), primitive_root))
   ....: 
Element:  1, Span: {1}          , Primitive root: False
Element:  2, Span: {8, 1, 2, 4} , Primitive root: False
Element:  4, Span: {1, 4}       , Primitive root: False
Element:  7, Span: {1, 4, 13, 7}, Primitive root: False
Element:  8, Span: {8, 1, 2, 4} , Primitive root: False
Element: 11, Span: {1, 11}      , Primitive root: False
Element: 13, Span: {1, 4, 13, 7}, Primitive root: False
Element: 14, Span: {1, 14}      , Primitive root: False

# Find the smallest primitive root
In [16]: galois.primitive_root(n)

# Find all primitive roots
In [17]: roots = galois.primitive_roots(n); roots
Out[17]: []

# Note the max order of any element is 4, not 8, which is Carmichael's lambda function
In [18]: galois.carmichael_lambda(n)
Out[18]: 4
Last update: Apr 03, 2022