counting the number of prime numbers up to some large value

Question

i wrote this super simple python code to count the number of prime numbers up to some large value. The problem is for a value like 1e+8the program takes so much time, i want to improve this code for faster result and better performance. here's the code:

import math

def is_prime(num):
    if num%2 ==0 and num>2:
        return 0
    for i in range(3, int(math.sqrt(num)) + 1, 2):
        if num % i == 0:
            return 0
    return 1

def count_prime(num):
    ct=0;
    for i in range(2,num+1,1):
        if is_prime(i)==1:
            ct+=1
    return ct

you can restrict the search for `i` to values in the form of 6k±1 where k is any integer — fferri, Aug 21 '18 at 12:27
This would be much better answered on codereview.stackexchange.com. — Martin Bonner supports Monica, Aug 21 '18 at 12:27
Try [**Sieve of Eratosthenes**](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes) — Ishpreet, Aug 21 '18 at 12:28
As @L3viathan says: If `is_prime` declares the number to be prime, add it to a (global) list. `is_prime` then only needs to check if `num` is divisible by the elements of `primes`. This will be *many* fewer checks. — Martin Bonner supports Monica, Aug 21 '18 at 12:28
Answered [here](https://stackoverflow.com/a/19072704/448810). — user448810, Aug 21 '18 at 12:32
I agree with L3viathan, but only for practical reasons. The people of earth have already computed this data upto staggering numbers, and lookup will trump computation. Though, the Sieve is always worth study :) — Davesoft, Aug 21 '18 at 12:32

score -1 · Answer 1 · answered Aug 21 '18 at 12:56

As suggested in the comments, you could start improving your is_prime() function. Try other algorithms like the suggested sieve, or the Miller Rabin test. I implemented the latter during a cryptography seminar and it wasnt to hard. (Note: the algorithm is a probabilistic algorithm, but dont be bothered by that to much). This could leed to a better time complexity for your prime check.

After this you could maybe think about multiple runs of ur count_prime() function. Here you could store the values from your runs and start from this point on the next start.

Example:

In the first run you calculate all primes for n = 5000. You get some result = x you store x for 5000.
In the second run maybe you enter 10.000 as ur designated input. Now you dont need to start over from 2 again, but you can start with your stored result and continue calculation from 5001. The addition of the results then is your result for 10.000

counting the number of prime numbers up to some large value

1 Answers1