import math

def sieveOfAtkin(numberToNotExceed):

    listOfPrimes = [2, 3, 5]
    sieve = [False] * (numberToNotExceed + 1)
    integerRootOfNumberToNotExceed = int(math.sqrt(numberToNotExceed)) + 1

    for x in range(1, integerRootOfNumberToNotExceed):
        for y in range(1, integerRootOfNumberToNotExceed):
            entryNumber = 4 * x ** 2 + y ** 2   ###   n = 4x^2 + y^2

            if (entryNumber <= numberToNotExceed) and (entryNumber % 12 == 1 or entryNumber % 12 == 5):
                sieve[entryNumber] = not sieve[entryNumber]

            entryNumber = 3 * x ** 2 + y ** 2   ###   n = 3x^2 + y^2

            if (entryNumber <= numberToNotExceed) and (entryNumber % 12 == 7):
                sieve[entryNumber] = not sieve[entryNumber]

            if x > y:
                entryNumber = 3 * x ** 2 - y ** 2   ###   n = 3x^2 - y^2

                if (entryNumber <= numberToNotExceed) and (entryNumber % 12 == 11):
                    sieve[entryNumber] = not sieve[entryNumber]


    for index in range(5, integerRootOfNumberToNotExceed):
        if sieve[index]:
            for composite in range(index ** 2, numberToNotExceed, index ** 2):
                sieve[composite] = False


    for index in range(7, numberToNotExceed):
        if sieve[index]:
            listOfPrimes.append(index)

    return listOfPrimes