Пример #1
0
def is_primitive_root(a, p):
    """
    Returns True if ``a`` is a primitive root of ``p``

    ``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and
    totient(p) is the smallest positive number s.t.

        a**totient(p) cong 1 mod(p)

    Examples
    ========

    >>> from sympy.ntheory import is_primitive_root, n_order, totient
    >>> is_primitive_root(3, 10)
    True
    >>> is_primitive_root(9, 10)
    False
    >>> n_order(3, 10) == totient(10)
    True
    >>> n_order(9, 10) == totient(10)
    False

    """
    a, p = int_tested(a, p)
    if igcd(a, p) != 1:
        raise ValueError("The two numbers should be relatively prime")
    if a > p:
        a = a % p
    if n_order(a, p) == totient(p):
        return True
    else:
        return False
Пример #2
0
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