Trying to calculate large numbers in Python with gmpy. Python keeps crashing?

I was recommended to use gmpy to assist with calculating large numbers efficiently. Before I was just using python and my script ran for a day or two and then ran out of memory (not sure how that happened because my program's memory usage should basically be constant throughout.. maybe a memory leak?)

Anyways, I keep getting this weird error after running my program for a couple seconds:

mp_allocate< 545275904->545275904 >
Fatal Python error: mp_allocate failure

This application has requested the Runtime to terminate it in an unusual way. 
Please contact the application's support team for more information.

Also, python crashes and Windows 7 gives me the generic python.exe has stopped working dialog.

This wasn't happening with using standard python integers. Now that I switch to gmpy I am getting this error just seconds in to running my script. I thought gmpy was specialized 开发者_如何学Goin dealing with large number arithmetic?

For reference, here is a sample program that produces the error:

import gmpy2

p = gmpy2.xmpz(3000000000)
s = gmpy2.xmpz(2)
M = s**p

for x in range(p):
    s = (s * s) % M

I have 10 gigs of RAM and without gmpy this script ran for days without running out of memory (still not sure how that happened considering s never really gets larger..

Anyone have any ideas?

EDIT: Forgot to mention I am using Python 3.2

Disclaimer: I'm the maintainer of gmpy and gmpy2.

I won't be able to test this until tonight. But a couple of comments and questions.

Instead of using (s * s) % M, use pow(s, 2, M). It should be faster.

What happens if you use gmpy2.mpz() instead of gmpy2.xmpz() ?

Are you running a 64-bit version of Python and gmpy2? (I assume so, but I just want to confirm.)

Regarding range vs. xrange, in Python 3.x, range has replaced xrange.

Edit with additional information.

The cause of the crash was due to overflow of internal structures in a 32-bit build. Using a 64-bit version of Python and gmpy or gmpy2 is the proper fix.

The unpack(x,n) function is similar to split() for a string: it divides a number into a series of n-bit values. It is equivalent to, but much faster than:

def unpack(x,n):
r = []
m = 2**n
while x:
    x, temp = divmod(x,m)
    r.append(temp)
return r

Some documentation is available via help(gmpy2.unpack) but better documentaion is on my to-do list.

The reason that unpack() can be used to eliminate the % operation is the same as the adding the digits of base-10 number to check for divisibility for 9. In this case, unpack() creates p-bit numbers and we are dividing by 2**p - 1.

Here is some test code:

import gmpy2
import time

def mersenne1(p):
    '''Primality test for Mersenne prime: 2**p -1.
    Uses native Python longs. Does not verify that p is prime.'''

    s = 4
    M = 2**p - 1
    for i in range(p-2):
        s = ((s*s)-2) % M
    return False if s else True

def mersenne2(p):
    '''Primality test for Mersenne prime: 2**p -1.
    Uses gmpy2.mpz. Does not verify that p is prime.'''

    s = gmpy2.mpz(4)
    M = gmpy2.mpz(2)**p - 1
    for i in range(p-2):
        s = ((s*s)-2) % M
    return False if s else True

def mersenne3(p):
    '''Primality test for Mersenne prime: 2**p -1.
    Uses gmpy2.mpz and no mod. Does not verify that p is prime.'''

    s = gmpy2.mpz(4)
    M = gmpy2.mpz(2)**p - 1
    for i in range(p-2):
        s = (s*s)
        s = sum(gmpy2.unpack(s, p))
        s = sum(gmpy2.unpack(s, p))
        if s < 2:
            s = M - 2 + s
        else:
            s = s - 2
    return False if s else True

if __name__ == "__main__":
    p = 44497

    start = time.time()
    result1 = mersenne1(p)
    print("Elapsed time: {:6.3f}".format(time.time() - start))

    start = time.time()
    result2 = mersenne2(p)
    print("Elapsed time: {:6.3f}".format(time.time() - start))

    start = time.time()
    result3 = mersenne3(p)
    print("Elapsed time: {:6.3f}".format(time.time() - start))

    if result1 == result2 == result3:
        print("All three tests are equal!")
    else:
        print("Oops, something has gone wrong.")

And some running times...

C:\x64\Python32>python.exe mersenne.py
Elapsed time: 163.683
Elapsed time: 12.782
Elapsed time:  3.630
All three tests are equal!