Checking divisibility for (sort of) big numbers in python

Question

I've been writing a simple program in python that encodes a string into a number using Gödel's encoding. Here's a quick overview: you take the first letter of the string, find its position in the alphabet (a -> 1, b -> 2, ..., z -> 26) and raise the first prime number (2) to this power. The you take the second letter in the string and the second prime (3) and so on. Here's the code:

import string, math
alphabet = list(string.ascii_lowercase)

def primes(n):
    "Returns a list of primes up to n."

    primes = [2, 3]
    i = 5
    while i < n:
        l = math.ceil(math.sqrt(i))
        k = math.ceil(math.sqrt(i+2))
        for p in primes[:l]:
            if i % p == 0:
                break
        else:
            primes.append(i)
        for p in primes[:k]:
            if (i+2) % p == 0:
                break
        else:
            primes.append(i+2)
        i += 6
    return primes

def Encode(string):
    "Encodes a string using Godel's encoding."

    enc = 1
    p = primes(100)
    for i in range(len(string)):
        enc = enc*(p[i]**(alphabet.index(string[i])+1))
    return enc

def Decode(code):
    "Decodes a Godel's encoding into a string."

    dec = ""
    for i in primes(100):
        count = 0
        while code % i == 0:
            code /= i
            count += 1
        if count == 0: #If we've found a prime that doesn't divide code, 
                       #there are no more letters to be added.
            break
        else:
            dec += alphabet[count-1]
    return dec

The primes() function works for my intends and purposes and so does Encode(). Now Decode() is the interesting part. It works for encodings up to ~15 digits long but starts doing some mystical stuff starting at ~20 digits. So for instance it gives the right output for the encoding of "aaaaaaaaaaaaaa" but not for "python". For big numbers it seems to execute the while code % i == 0 loop too many times (176 for the first letter of "python" when it should be just 16).

Is this just a problem with the mod function in python? Sounds strange as 20 digits isn't all that long for a computer. Is there a mistake in my code? Thanks for all the help. I'm not a programmer myself but I'm trying to learn doing stuff like this. Therefore any constructive criticism is welcome.

Answer 1

/= in Python 3 returns a double-precision floating point value. (As does math.ceil, btw.) Floating point values do not have arbitrary precision. You could use //= instead. That always results in an integer. (It gives the floor of the result.)

(I previously said that math.ceil was your main culprit. I don't think that's the case, but nonetheless, you probably shouldn't be using a floating point value to index into a list. If you need to run the same code in Python 2 that will fail. You can cast it back to an integer using int(math.ceil(...)), although you might want to consider avoiding floating-point calculations altogether, since things will begin to break down for sufficiently large values.)