大量的Python优化

Question

I want to print all prime numbers with 7 consecutive 7s less than 10000000000 . I was getting a MemoryError when using range() because the generated array couldn't be stored, so I changed the loop into a while loop.

However the program is really slow. It takes more than a minute to print the first number found.

import math

def is_prime(n):
    if n % 2 == 0 and n > 2: 
        return False
    i = 3
    while i < math.sqrt(n) + 1:
    #for i in range(3, int(math.sqrt(n)) + 1, 2):
        if n % i == 0:
            return False
        i += 2
    return True

def is_super_happy(n):
    count = 0
    while n != 0:
        if n%10 == 7:
            count += 1
            if count == 7:
                return True
        else:
            count = 0
        n /= 10
    return count == 7

i = 7777777
while i < 10e10:
#for i in range(7777777, int(10e10)):
    if is_super_happy(i) and is_prime(i):
        print i
    i += 1

I can't thing of anything that could make this go faster, and I want it to be really fast.

Any ideas, tips?

Answer 1

Your current algorithm is very wasteful: is_prime checks every number from 1 to sqrt(n) on every iteration, whereas it should only check known prime numbers.

Modify your algorithm so that it implements http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

In addition, you're computing sqrt(n) at every tick of your while loop (potentially several million times if you get high enough), except that it doesn't change. Compute it once at the beginning of the function and reuse that.

Answer 2

Two suggestions:

• Generate the 7, 8 and 9-digit numbers less than 10e10 containing 7777777 directly (rather than checking each and every number in turn) and then check them. There are a relatively small number of such numbers.

  For example, here's a way to generate a list of all such numbers which end with '7777777': [int(str(x)+'7777777') for x in xrange(100)]

• Consider using a probabilistic test for checking primality (such as Miller-Rabin ). This tells you whether a number x is prime with a very high level of probability and is far quicker that checking potentially thousands of numbers smaller than x for divisibility.

Answer 3

I find 250 primes in about a quarter second, I don't want to give my code because your question looks like homework.

Methods:

• first, find all the primes below 100 000 (that's sqrt(your max number)), using a sieve.
• is_prime() only needs to check whether any of those primes divide the number
• There are only 4000 of these numbers; consider the numbers 000 to 999. They can be combined with 7777777 in four ways; say 123 becomes 7777777123, 1777777723, 1277777773 and 1237777777. Generate those, check them against primality, done.
• Use strings ( int("12" + "7777777" + "3") ) or just math (12 * 100000000 + 7777777 * 10 + 3) to generate the numbers