def _unit_gens_primepowercase(p, r):
r"""
Return a list of generators for `(\ZZ/p^r\ZZ)^*` and their orders.
EXAMPLES::
sage: from sage.rings.finite_rings.integer_mod_ring import _unit_gens_primepowercase
sage: _unit_gens_primepowercase(2, 3)
[(7, 2), (5, 2)]
sage: _unit_gens_primepowercase(17, 1)
[(3, 16)]
sage: _unit_gens_primepowercase(3, 3)
[(2, 18)]
"""
pr = p**r
if p == 2:
if r == 1:
return []
if r == 2:
return [(integer_mod.Mod(3, 4), integer.Integer(2))]
return [(integer_mod.Mod(-1, pr), integer.Integer(2)),
(integer_mod.Mod(5, pr), integer.Integer(2**(r - 2)))]
# odd prime
return [(integer_mod.Mod(primitive_root(pr, check=False),
pr), integer.Integer(p**(r - 1) * (p - 1)))]
def _unit_gens_primepowercase(p, r):
r"""
Return a list of generators for `(\ZZ/p^r\ZZ)^*` and their orders.
EXAMPLES::
sage: from sage.rings.finite_rings.integer_mod_ring import _unit_gens_primepowercase
sage: _unit_gens_primepowercase(2, 3)
[(7, 2), (5, 2)]
sage: _unit_gens_primepowercase(17, 1)
[(3, 16)]
sage: _unit_gens_primepowercase(3, 3)
[(2, 18)]
"""
pr = p**r
if p == 2:
if r == 1:
return []
if r == 2:
return [(integer_mod.Mod(3, 4), integer.Integer(2))]
return [(integer_mod.Mod(-1, pr), integer.Integer(2)),
(integer_mod.Mod(5, pr), integer.Integer(2**(r - 2)))]
# odd prime
return [(integer_mod.Mod(primitive_root(pr, check=False), pr),
integer.Integer(p**(r - 1) * (p - 1)))]