Is there a simple algorithm that can determine if X is prime?

Question

I have been trying to work my way through Project Euler, and have noticed a handful of problems ask for you to determine a prime number as part of it.

1. I kno

Answer 1

Here's a simple optimization of your method that isn't quite the Sieve of Eratosthenes but is very easy to implement: first try dividing X by 2 and 3, then loop over j=1..sqrt(X)/6, trying to divide by 6*j-1 and 6*j+1. This automatically skips over all numbers divisible by 2 or 3, gaining you a pretty nice constant factor acceleration.

Answer 2

Your right the simples is the slowest. You can optimize it somewhat.

Look into using modulus instead of square roots. Keep track of your primes. you only need to divide 7 by 2, 3, and 5 since 6 is a multiple of 2 and 3, and 4 is a multiple of 2.

Rslite mentioned the eranthenos sieve. It is fairly straight forward. I have it in several languages it home. Add a comment if you want me to post that code later.

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Here is my C++ one. It has plenty of room to improve, but it is fast compared to the dynamic languages versions.

// Author: James J. Carman
// Project: Sieve of Eratosthenes
// Description: I take an array of 2 ... max values. Instead of removeing the non prime numbers,
// I mark them as 0, and ignoring them.
// More info: http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
#include <iostream>

int main(void) {
        // using unsigned short.
        // maximum value is around 65000
        const unsigned short max = 50000;
        unsigned short x[max];
        for(unsigned short i = 0; i < max; i++)
                x[i] = i + 2;

        for(unsigned short outer = 0; outer < max; outer++) {
                if( x[outer] == 0)
                        continue;
                unsigned short item = x[outer];
                for(unsigned short multiplier = 2; (multiplier * item) < x[max - 1]; multiplier++) {
                        unsigned int searchvalue = item * multiplier;
                        unsigned int maxValue = max + 1;
                        for( unsigned short maxIndex = max - 1; maxIndex > 0; maxIndex--) {
                                if(x[maxIndex] != 0) {
                                        maxValue = x[maxIndex];
                                        break;
                                }
                        }
                        for(unsigned short searchindex = multiplier; searchindex < max; searchindex++) {
                                if( searchvalue > maxValue )
                                        break;
                                if( x[searchindex] == searchvalue ) {
                                        x[searchindex] = 0;
                                        break;
                                }
                        }
                }
        }
        for(unsigned short printindex = 0; printindex < max; printindex++) {
                if(x[printindex] != 0)
                        std::cout << x[printindex] << "\t";
        }
        return 0;
}

I will throw up the Perl and python code I have as well as soon as I find it. They are similar in style, just less lines.

Answer 3

I like this python code.

def primes(limit) :
    limit += 1
    x = range(limit)
    for i in xrange(2,limit) :
        if x[i] ==  i:
            x[i] = 1
            for j in xrange(i*i, limit, i) :
                x[j] = i
    return [j for j in xrange(2, limit) if x[j] == 1]

A variant of this can be used to generate the factors of a number.

def factors(limit) :
    limit += 1
    x = range(limit)
    for i in xrange(2,limit) :
        if x[i] == i:
            x[i] = 1
            for j in xrange(i*i, limit, i) :
                x[j] = i
    result = []
    y = limit-1
    while x[y] != 1 :
        divisor = x[y]
        result.append(divisor)
        y /= divisor
    result.append(y)
    return result

Of course, if I were factoring a batch of numbers, I would not recalculate the cache; I'd do it once and do lookups in it.

Answer 4

For reasonably small numbers, x%n for up to sqrt(x) is awfully fast and easy to code.

Simple improvements:

test 2 and odd numbers only.

test 2, 3, and multiples of 6 + or - 1 (all primes other than 2 or 3 are multiples of 6 +/- 1, so you're essentially just skipping all even numbers and all multiples of 3

test only prime numbers (requires calculating or storing all primes up to sqrt(x))

You can use the sieve method to quickly generate a list of all primes up to some arbitrary limit, but it tends to be memory intensive. You can use the multiples of 6 trick to reduce memory usage down to 1/3 of a bit per number.

I wrote a simple prime class (C#) that uses two bitfields for multiples of 6+1 and multiples of 6-1, then does a simple lookup... and if the number i'm testing is outside the bounds of the sieve, then it falls back on testing by 2, 3, and multiples of 6 +/- 1. I found that generating a large sieve actually takes more time than calculating primes on the fly for most of the euler problems i've solved so far. KISS principle strikes again!

I wrote a prime class that uses a sieve to pre-calculate smaller primes, then relies on testing by 2, 3, and multiples of six +/- 1 for ones outside the range of the sieve.