# Finding lucas pseudoprimes
from math import sqrt, gcd

def isqrt(n):
    """ Find the integer square root of n via newton's method, code via
    stackoverflow:

    (https://stackoverflow.com/questions/15390807/integer-square-root-in-python)
    """
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

def hasIntSQRT(n):
    """Detect whether the square root of n is an integer,
    i.e. whether the isqrt(n) is the true square root.
    """
    isq = isqrt(n)
    return isq * isq == n

def Dsequence():
    """Generate sequence 5, -7, 9, -11, 13, -15...
    """
    val = 5
    while True:
        if val % 4 == 1:
            yield val
        else:
            yield -val
        val = val + 2

def Legendre(a, p):
    """Function for calculating Legendre symbol.

    Note that this is only supposed to be defined if p is an odd prime, but we
    don't know in advance if that is true - will therefore only throw and error
    if p is even.

    Uses original power definition from
    <https://en.wikipedia.org/wiki/Legendre_symbol>
    """
    if (p % 2 == 0):
        raise ValueError("p must be odd, is {}".format(p))
    lv =  pow(a, (p-1)//2, p)
    if lv == p - 1:
        lv = -1
    return lv

def Jacobi(a, n):
    """Function for calculating Jacobi symbol.

    Note that this is only defined for positive odd integers n.

    Uses algorithm from
    <https://en.wikipedia.org/wiki/Jacobi_symbol#Calculating_the_Jacobi_symbol>
    """
    if n < 1:
        raise ValueError("n must be positive")
    if (n % 2 == 0):
        raise ValueError("n must be odd")
    if a % n != a:
        return Jacobi(a%n, n)
    if a != 0 and a % 2 == 0:
        nm8 = n % 8
        if (nm8 == 3 or nm8 == 5):
            return -1 * Jacobi(a//2, n)
        else:
            return Jacobi(a//2, n)
    if a == 1:
        return 1
    if gcd(a, n) != 1:
        return 0
    if a == 0 and n == 1:
        return 1
    if n % 4 == 3 and a % 4 == 3:
        return -1 * Jacobi(n, a)
    return Jacobi(n, a)

print("   ", end=" ")
for k in range(1, 10):
    print("{}".format(k).rjust(2, " "), end=" ")
print()

for n in range(1, 21, 2):
    print("{}:".format(n).rjust(3, " "), end=" ")
    for k in range(1, 10):
        print("{}".format(Jacobi(k, n)).rjust(2, " "), end=" ")
    print()