def n_order(a, n):
"""Returns the order of ``a`` modulo ``n``.
The order of ``a`` modulo ``n`` is the smallest integer
``k`` such that ``a**k`` leaves a remainder of 1 with ``n``.
Examples
========
>>> from sympy.ntheory import n_order
>>> n_order(3, 7)
6
>>> n_order(4, 7)
3
"""
a, n = int_tested(a, n)
if igcd(a, n) != 1:
raise ValueError("The two numbers should be relatively prime")
group_order = totient(n)
factors = factorint(group_order)
order = 1
if a > n:
a = a % n
for p, e in factors.iteritems():
exponent = group_order
for f in xrange(0, e + 1):
if (a ** (exponent)) % n != 1:
order *= p ** (e - f + 1)
break
exponent = exponent // p
return order
def n_order(a, n):
""" returns the order of a modulo n
Order of a modulo n is the smallest integer
k such that a^k leaves a remainder of 1 with n.
"""
if igcd(a, n) != 1:
raise ValueError("The two numbers should be relatively prime")
group_order = totient_(n)
factors = factorint(group_order)
order = 1
if a > n:
a = a % n
for p, e in factors.iteritems():
exponent = group_order
for f in xrange(0, e + 1):
if (a ** (exponent)) % n != 1:
order *= p ** (e - f + 1)
break
exponent = exponent // p
return order
def n_order(a, n):
""" returns the order of a modulo n
Order of a modulo n is the smallest integer
k such that a^k leaves a remainder of 1 with n.
"""
if igcd(a, n) != 1:
raise ValueError("The two numbers should be relatively prime")
group_order = totient_(n)
factors = factorint(group_order)
order = 1
if a > n:
a = a % n
for p, e in factors.iteritems():
exponent = group_order
for f in xrange(0, e + 1):
if (a**(exponent)) % n != 1:
order *= p**(e - f + 1)
break
exponent = exponent // p
return order
