Simple Mersenne prime search

For fun I tried to search for Mersenne primes with a very simple code (below). It finds the 20 first Mersenne primes (all known by 1961) in under a minute. After that it slows down quite a lot and finds the 26th prime (discovered in 1979) in about 5 hours.

mersenne_primes

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import time
import math
 
# http://stackoverflow.com/questions/16004407/a-fast-prime-number-sieve-in-python
def prime_sieve(n):
    size = n//2
    sieve = [1]*size
    limit = int(n**0.5)
    for i in range(1,limit):
        if sieve[i]:
            val = 2*i+1
            tmp = ((size-1) - i)//val 
            sieve[i+val::val] = [0]*tmp
    return [2] + [i*2+1 for i, v in enumerate(sieve) if v and i>0]
 
# https://en.wikipedia.org/wiki/Trial_division
def trial_division(n):
    """Return a list of the prime factors for a natural number."""
    if n < 2:
        return []
    prime_factors = []
    for p in prime_sieve(int(n**0.5) + 1):
        if p*p > n: break
        while n % p == 0:
            prime_factors.append(p)
            n //= p
    if n > 1:
        prime_factors.append(n)
    return prime_factors
 
def is_prime_trial_division(n):
    return len(trial_division(n)) == 1
 
# https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test
def is_prime_lucas_lehmer(p):
    s = 4
    M = pow(2,p) -1
    for n in range(p-2):
        s = (pow(s,2)-2) % M
    return s==0
 
n=2 
t0 = time.time()
print "#\tp\ttime\tyear"
log_max_p=15
for p in range(1,pow(2,log_max_p)):
    if is_prime_trial_division(p):
        if is_prime_lucas_lehmer(p):
            elapsed = time.time() - t0
            print "%4d\t%4d\t%.1f s" % (n, p, elapsed)
            n=n+1

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