Python script for search for factors of M1277 using random k-intervals

[Viliam Furik]

import numpy as np
from random import randint

f = open("m1277.txt", "a")


def tf1277(k, r):
    expos = np.zeros((4, r), int)
    p = 1277
    y = 0
    z = 0
    e1 = 2555
    e2 = 3832
    e3 = 7663
    e4 = 8940
    pbinary = []
    q = p
    while q > 0:
        if q % 2 == 0:
            pbinary.append(0)
            q //= 2
        if q % 2 == 1:
            pbinary.append(1)
            q //= 2
    pbinary.reverse()
    for x in range(4):
        for prime in [3, 5, 7, 11, 13, 17, 19, 23, 29]:
            for a in range(prime + 5):
                if x == 0:
                    e = e1
                elif x == 1:
                    e = e2
                elif x == 2:
                    e = e3
                else:
                    e = e4
                if ((a + k) * 10216 + e) % prime == 0:
                    c = a
                    while c < r:
                        expos[x][c] = 1
                        c += prime
                    break
    for a in range(len(expos[0])):
        if expos[0][a] == 0:
            c = 1
            kp = (k + a) * 10216 + 2555
            for d in pbinary:
                c **= 2
                if d == 1:
                    c *= 2
                c %= kp
            if c == 1:
                print("-------------------------")
                print("factor = ", kp)
                print("k = ", (kp - 1) // 2 // 1277)
                f.write("\n")
                f.write("factor = " + str(kp))
                f.write("\n")
                f.write("k = " + str((kp - 1) // 2 // 1277))
                f.write("\n")
                quit()
            if y == 1000:
                y = 0
                z += 1
                print(z * 1000)
            y += 1
    for a in range(len(expos[1])):
        if expos[1][a] == 0:
            c = 1
            kp = (k + a) * 10216 + 3832
            for d in pbinary:
                c **= 2
                if d == 1:
                    c *= 2
                c %= kp
            if c == 1:
                print("-------------------------")
                print("factor = ", kp)
                print("k = ", (kp - 1) // 2 // 1277)
                f.write("\n")
                f.write("factor = " + str(kp))
                f.write("\n")
                f.write("k = " + str((kp - 1) // 2 // 1277))
                f.write("\n")
                quit()
            if y == 1000:
                y = 0
                z += 1
                print(z * 1000)
            y += 1
    for a in range(len(expos[2])):
        if expos[2][a] == 0:
            c = 1
            kp = (k + a) * 10216 + 7663
            for d in pbinary:
                c **= 2
                if d == 1:
                    c *= 2
                c %= kp
            if c == 1:
                print("-------------------------")
                print("factor = ", kp)
                print("k = ", (kp - 1) // 2 // 1277)
                f.write("\n")
                f.write("factor = " + str(kp))
                f.write("\n")
                f.write("k = " + str((kp - 1) // 2 // 1277))
                f.write("\n")
                quit()
            if y == 1000:
                y = 0
                z += 1
                print(z * 1000)
            y += 1
    for a in range(len(expos[3])):
        if expos[3][a] == 0:
            c = 1
            kp = (k + a) * 10216 + 8940
            for d in pbinary:
                c **= 2
                if d == 1:
                    c *= 2
                c %= kp
            if c == 1:
                print("-------------------------")
                print("factor = ", kp)
                print("k = ", (kp - 1) // 2 // 1277)
                f.write("\n")
                f.write("factor = " + str(kp))
                f.write("\n")
                f.write("k = " + str((kp - 1) // 2 // 1277))
                f.write("\n")
                quit()
            if y == 1000:
                y = 0
                z += 1
                print(z * 1000)
            y += 1


while True:
    tf1277(randint(58000000000000000, 2 ** (1277 / 2) / 1277 / 2), 1000000)

Hello, I have made the above Python script, that is called with random starting k value, and then tests about 1263000 sived-out k's above that value. It uses a grid of zeroes to sieve k's and combines the facts that factors must be 2*k*p+1 and also 1 or 7 (mod 8) - so it only tests factors that are 2555, 3832, 7663, 8940 (mod 10216).

If there is a mistake (either logical or mathematical) in my script, I'll be grateful for pointing out. I would also be grateful if somebody could help me with making the code runnable on a Nvidia GPU with CUDA (or in OpenCL, if it's easier), without necessarily using FFT or Barrett multiplication. (But if somebody is willing to explain, I'll be happy to listen, or read...).

And for the last, if you want, you can try to find the factor yourself with that code. It takes my CPU about 12 seconds on average to go through random range of size 10216000000 (that is difference between largest and smallest k tested). So if I was to randomly check the equivalent number of ranges (untested range from 2^67 to 2^639 divided by range size 10216000000), it would take me... about 2*10^172 years . But if more people engage in search, probability increases (unfortunately it's only slight change).

So if you want to help, you can run the script any time you like. If you want to help more, you can help with improving the speed of the script.

[retina]

Running this code is a fantastic way to waste power/time/money/energy.

Random TF for this number is pointless. And trying to "optimise" a scripting language is equally pointless.

You need another method (not TF), and you need another language (not a scripting language).

[mathwiz]

Originally Posted by Viliam Furik

if more people engage in search, probability increases (unfortunately it's only slight change).

So if you want to help, you can run the script any time you like. If you want to help more, you can help with improving the speed of the script.

You'd probably have a better chance of being struck by lightning 10 times and winning the lottery all on the same day.

The only hope for M1277, with presently known factoring techniques, is a mountain of ECM or (eventually) SNFS.

[Runtime Error]

Fun! I have a machine that is ram-bottle-necked for Prime95. So far, running this too doesn't seem to affect the iterations per second. I'll leave it on for a week or so and perhaps lightning with strike! (Or strike 10 times, per the comment above. But I will probably not buy the requisite lotto ticket.)

[VBCurtis]

How long will it take this program to search for all 70-digit factors? Or, most of them? How about 1% of the 70-digit range?

ECM on this number has ruled out factors smaller than about 67 digits, and a 70-digit or smaller factor is unlikely. So, I'm not even sure you should set it to look at 70 digits- maybe 73 is smarter. How many k's are there that make 73-digit candidate factors?

[firejuggler]

we are looking at a 245 bit factor? hmmm
66-67 bits TF took 11 704 Ghz/D, and it double each bit.
so 245-67=178. Sooo 2^178*11 704= 4.484e57 Ghz/D for the last bit. Not counting all that is before it,
Which double it. Lets handwave this a bit and it come to 9e57Ghz/D.

Problem: The fastest current TF card,will give you 6e3Ghz/D by day. so, AFAIK, it would take 1.5e54 Day,

About 3e41 time the duration since the beginning of the universe. Thats, by TF only. PM1 is almost excluded because of its requirement.

[VBCurtis]

So, one fast GPU can spend 1e54 days to have roughly a 1 in 250 chance to find a factor.

Or, you could run regular ECM with large bounds (B1=3e9 or bigger) and actually contribute non-useless work.

[Uncwilly]

Is there enough data yet to suggest that Mersenne non-primes have a higher likelihood to have 3 only factors vs 2 only?

[Batalov]

Originally Posted by Uncwilly

Is there enough data yet to suggest that Mersenne non-primes have a higher likelihood to have 3 only factors vs 2 only?

Many more Mersenne non-primes have a thousand factors. And even more have a million factors!
Infinity is a rather large thing.

[LaurV]

Originally Posted by firejuggler

Problem: The fastest current TF card,will give you 1.2e4Ghz/D by day.

Fixed it for you. Not that would change much, haha.

One-line pari "search by q" (as posted a lot of times in the forum) with a parfor and a random start would be faster that the posted script.

[Dr Sardonicus]

Originally Posted by Uncwilly

Is there enough data yet to suggest that Mersenne non-primes have a higher likelihood to have 3 only factors vs 2 only?

Exactly three versus exactly two prime factors, not sure. Exactly two versus more than two, long-known data suggest "no contest."

I'll assume, for the sake of discussion, that you mean Mersenne numbers M_p with prime exponent p. For composite exponents n, the algebraic factorization alone is very likely to provide more than two proper factors, which, in turn, are very likely to provide more than two prime factors.

From an old factorization table of 2ⁿ - 1 for odd n < 1200 I have on file, a quick scan finds that M_p was known to be prime for 13 odd prime p, known to have exactly two prime factors for 38 prime p, and known to have 3 or more prime factors for 143 odd prime p. The table lists M₁₀₆₁ as a C320 so the number of factors was only known to be 2 or more at the time.

It seems that 2¹⁰⁶¹ - 1 was subsequently found to be the product of two prime factors, so M_p is known to be prime for 13 odd prime p < 1200, known to have exactly two prime factors for 39 odd primes p < 1200, and known to have 3 or more prime factors for 143 odd prime p < 1200. That accounts for all 195 odd primes p < 1200.

So I would say that the data suggest that more than two factors is much more likely than exactly two.

There are only about 400 primes p for which the exact number of factors of 2^p - 1 is known or "probably" known. I'm too lazy to check for how many p the number of factors is known to be at least 3.