====Lösung der Aufgaben 1 und 2====

**Aufgabe 1**

a) falsche Aussage\\
b) keine Aussage\\
c) keine Aussage\\
d) keine Aussage\\
e) wahre Aussage\\
f) keine Aussage (Goldbachsche Vermutung)

**Aufgabe 2**

<code python goldbach.py>
def istPrimzahl(n):
    prim = True
    k = 2
    while k*k <= n and prim == True:
        if n%k == 0:
            prim = False
            break
        k +=1
    return prim

t = "J"
while t == "J":
    n = int(input("Gerade Zahl größer 2: "))
    while not(n > 2 and n%2 == 0):
        n = int(input("Gerade Zahl größer 2: "))

    a = 2
    b = n-2
    while a <= n//2:
        if istPrimzahl(a):
            if istPrimzahl(b):
                break
        a += 1
        b -= 1
    print(a,"+",b,"=",n)
    t = input("Nochmal? J/N: ")

</code>

<code python goldbach_miller_rabin.py>
import random

def istPrimzahl(n,k):

    # Implementation uses the Miller-Rabin Primality Test
    # The optimal number of rounds for this test is 40
    # See http://stackoverflow.com/questions/6325576/how-many-iterations-of-rabin-miller-should-i-use-for-cryptographic-safe-primes
    # for justification

    # If number is even, it's a composite number

    if n < 0:
        return False

    if n == 1:
        return False

    if n == 2 or n == 3:
        return True

    if n % 2 == 0:
        return False

    r, s = 0, n - 1
    while s % 2 == 0:
        r += 1
        s //= 2
    for _ in range(k):
        a = random.randrange(2, n - 1)
        x = pow(a, s, n)
        if x == 1 or x == n - 1:
            continue
        for _ in range(r - 1):
            x = pow(x, 2, n)
            if x == n - 1:
                break
        else:
            return False
    return True


t = "J"
while t == "J":
    n = int(input("Gerade Zahl größer 2: "))
    while not(n > 2 and n%2 == 0):
        n = int(input("Gerade Zahl größer 2: "))

    a = 2
    b = n-2
    while a <= n//2:
        if istPrimzahl(a,1):
            if istPrimzahl(b,1):
                break
        a += 1
        b -= 1
    print(a,"+",b,"=",n)
    t = input("Nochmal? J/N: ")

</code>