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Prime Number Counter in C

BuckGup
By BuckGup
in Programming
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Hello I was wondering what is the fastest algorithm to find the number of prime numbers between two given numbers? I see many brute force algorithms that use breaks but I want it to be as fast as possible and not use any breaks. Thanks

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Just keeping this here as a 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Counting them gets computationally difficult very fast. If it is sufficient to have an approximation, see link below where you can get an estimate of number of primes below a certain number. Work it out for the two numbers of interest and subtract accordingly.

https://primes.utm.edu/howmany.html

 

It was pointed out to me that logs as used are natural logs, not to be confused with log base 10.

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57 minutes ago, BuckGup said:

I see many brute force algorithms that use breaks but I want it to be as fast as possible and not use any breaks.

Are you referring to a break-statement, e.g. inside a loop or inside a switch-statement? This suggests the algorithms in question short-circuit in order to improve performance. A break-statement as such is not an indicator of bad performance.

 

To find prime numbers you basically need to test whether you can divide the number in question by anything other than itself or 1. There are certain performance improvements you can do, e.g. you only need to test division by previously found prime numbers and you only need to test numbers that are at most half as large as the number you're testing, but overall finding prime numbers is very compute intensive.

 

Take a look at Prime95 and GIMPS, these projects have been running for years on a huge number of computers to find ever larger prime numbers.

Remember to either quote or @mention others, so they are notified of your reply

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5 hours ago, BuckGup said:

Hello I was wondering what is the fastest algorithm to find the number of prime numbers between two given numbers? I see many brute force algorithms that use breaks but I want it to be as fast as possible and not use any breaks. Thanks

This is very important: Do you need to know exactly how many primes are between two numbers a and b, or do you need to know roughly how many primes there are between a and b?

Additionally, what is the largest value that you expect to have to find primes under? Stated differently, given that pi(x) returns the number of primes below some number x, and given that pi(b) - pi(a) gives the number of primes between a and b, what is the largest x that pi(x) needs to handle?

I ask because those things can have a large impact on which algorithms can be used, especially if you don't need an exact count. If you can handle a close estimate, then there are closed form solutions available.
 

ENCRYPTION IS NOT A CRIME

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An alternative option would be to utilize a precalculated list of primes and simply turn this into a search/sort optimization problem. More storage/memory intensive but potentially the fastest solution.

 

 

https://primes.utm.edu/lists/small/millions/

 

 

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You'd have to work them out sequentially. There are a lot of good algorithms for this, including the sieve of Eratosthenes. You could also include a list of precalculated primes to speed up the process with higher numbers.

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5 hours ago, Eigenvektor said:

Are you referring to a break-statement, e.g. inside a loop or inside a switch-statement? This suggests the algorithms in question short-circuit in order to improve performance. A break-statement as such is not an indicator of bad performance.

 

To find prime numbers you basically need to test whether you can divide the number in question by anything other than itself or 1. There are certain performance improvements you can do, e.g. you only need to test division by previously found prime numbers and you only need to test numbers that are at most half as large as the number you're testing, but overall finding prime numbers is very compute intensive.

 

Take a look at Prime95 and GIMPS, these projects have been running for years on a huge number of computers to find ever larger prime numbers.

 

4 hours ago, straight_stewie said:

This is very important: Do you need to know exactly how many primes are between two numbers a and b, or do you need to know roughly how many primes there are between a and b?

Additionally, what is the largest value that you expect to have to find primes under? Stated differently, given that pi(x) returns the number of primes below some number x, and given that pi(b) - pi(a) gives the number of primes between a and b, what is the largest x that pi(x) needs to handle?

I ask because those things can have a large impact on which algorithms can be used, especially if you don't need an exact count. If you can handle a close estimate, then there are closed form solutions available.
 

Yes I need to know exactly how many. The largest value is 1 million.

3 hours ago, BlueJedi said:

An alternative option would be to utilize a precalculated list of primes and simply turn this into a search/sort optimization problem. More storage/memory intensive but potentially the fastest solution.

 

 

https://primes.utm.edu/lists/small/millions/

 

 

 

3 hours ago, Sauron said:

You'd have to work them out sequentially. There are a lot of good algorithms for this, including the sieve of Eratosthenes. You could also include a list of precalculated primes to speed up the process with higher numbers.

Will look into that

▓ƆԀ▓ S▓₱▓Ɇ▓c▓s▓: i7 6ʇɥפᴉƎ00K (4.4ghz), Asus DeLuxe X99A II, GT҉X҉1҉0҉8҉0 Zotac Amp ExTrꍟꎭe),Si6F4Gb D???????r PlatinUm, EVGA G2 Sǝʌǝᘉ5ᙣᙍᖇᓎᙎᗅᖶt, Phanteks Enthoo Primo, 3TB WD Black, 500gb 850 Evo, H100iGeeTeeX, Windows 10, K70 R̸̢̡̭͍͕̱̭̟̩̀̀̃́̃͒̈́̈́͑̑́̆͘͜ͅG̶̦̬͊́B̸͈̝̖͗̈́, G502, HyperX Cloud 2s, Asus MX34. פN∩SW∀S 960 EVO

Just keeping this here as a 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Here's some code in Java that includes some optimizations (I'm sure there's more to be had). It calculates the prime numbers up to 1 million. It takes about 100ms to calculate (with Java VM startup overhead), so fast enough in my book :)

 

Sorry, my C is a little rusty, but it should be simple enough to port. With that list you could then easily find the primes that are larger than your lower boundary and smaller than your upper boundary. 

private boolean isPrimeNumber(final int value, final List<Integer> primes) {
	// Divisors larger than the square root of 'value' don't need to be tested
	final double root = Math.sqrt(value);

	for (final Integer prime : primes) {
		// If no divisor found by now, number is a prime
		if (prime > root) {
			return true;
		}

		// If 'value' is evenly divisible by 'prime' it is not a prime
		if (value % prime == 0) {
			return false;
		}
	}
	throw new IllegalStateException("This should never happen ;)");
}

public void primeNumberGenerator() {
	final List<Integer> primes = new ArrayList<>();
	// Skip '2' because we don't test even numbers
	primes.add(3);
	primes.add(5);

	for (int value = 7; value < 1_000_000; value += 2) {
		if (isPrimeNumber(value, primes)) {
			primes.add(value);
		}
	}

	// Add '2' as first prime number
	primes.add(0, 2);

	// Some quick sanity checks
	assert primes.size() == 78_498 : "Unexpected number of prime numbers below 1,000,000 found";
	assert primes.get(3) == 7;
	assert primes.get(25) == 101;
	assert primes.get(299) == 1_987;
	assert primes.get(979) == 7_723;
	assert primes.get(18_000) == 200_191;
	assert primes.get(45_000) == 545_749;
	assert primes.get(78_497) == 999_983;

	// Show result
	System.out.println(StringUtils.join(primes, ","));
}

 

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On 11/9/2019 at 12:17 AM, BuckGup said:

Yes I need to know exactly how many. The largest value is 1 million.

If you don't mind me asking, I am curious what your use case is with these?

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On 11/9/2019 at 12:17 AM, BuckGup said:

 

Yes I need to know exactly how many. The largest value is 1 million.

 

Will look into that

Finding the number of primes between 1 and 1,000,000 would take at most a few seconds on a modern computer with the simple approach. That's based on the calculation that at most you need to test sqrt(k) divisions per number, and ∑_k=1^1000000 k^0.5 = 6.7x10^8. Modern cpus have a clock speed of the order 4x10^9 Hz, although each division takes several cycles and you will have other operations too, so they have about the same order of magnitude. 

 

Of course, I only did this on paper, so I've not tried it, but if a few seconds is good enough then it's worth a try. 

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1 hour ago, colonel_mortis said:

Of course, I only did this on paper, so I've not tried it, but if a few seconds is good enough then it's worth a try. 

Take a look at my post above. My mostly unoptimized Java code above takes ~100ms, and that includes start up of the VM. If you needed (a lot) more primes, I would probably use multi-threading to further improve the speed.

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Reading this I figured I'd write a naive multi-threaded one with Go as an exercise for channels, because well, eh, why not. Thought it's pretty nice how easy it is to make all cores work. Since it uses channels, it can't accommodate a proper solution, you'd need a shared memory prime number collection with locks, too lazy for that shiz. Though, it can work on any interval you choose.

Spoiler 

package main

import (
	"fmt"
	"runtime"
	"sync"
	"time"
)

// NumberGenerator generate numbers for processing
func NumberGenerator(lower, upper uint) <-chan uint {
	numChannel := make(chan uint, runtime.NumCPU()*10)
	go func() {
		for idx := lower; idx <= upper; idx++ {
			if idx%2 == 0 {
				continue
			}
			numChannel <- idx
		}
		close(numChannel)
	}()
	return numChannel
}

// PrimeProcessor concurrenly checks numbers in channel and puts prime numbers to output channel
func PrimeProcessor(numChan <-chan uint, output chan<- uint, wg *sync.WaitGroup) {
	for idx := 0; idx < runtime.NumCPU(); idx++ {
		wg.Add(1)
		go func() {
			for num := range numChan {
				if IsPrime(num) {
					output <- num
				}
			}
			wg.Done()
		}()
	}
}

// IsPrime checks if the number is a prime number
func IsPrime(num uint) bool {
	checkingBound := num / 3
	for divisor := uint(3); divisor <= checkingBound; divisor += 2 {
		if num%divisor == 0 {
			return false
		}
	}
	return true
}

// PrimeNumbers returns an array of prime numbers that exist within the given bounds
func PrimeNumbers(lower, upper uint) []uint {
	resultChannel := make(chan uint, runtime.NumCPU()*10)
	numChannel := NumberGenerator(lower, upper)

	results := make([]uint, 0, (upper-lower)/1000)
	go func() {
		for result := range resultChannel {
			results = append(results, result)
		}
	}()

	var wg sync.WaitGroup
	PrimeProcessor(numChannel, resultChannel, &wg)
	wg.Wait()

	close(resultChannel)
	return results
}

func main() {
	start := time.Now()
	results := PrimeNumbers(0, 1000000)
	finish := time.Now()
	for _, n := range results {
		fmt.Printf("%d\n", n)
	}
	fmt.Printf("Time spent calculating primes: %d ns\n", finish.Sub(start).Nanoseconds())
}

 

 

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On 11/8/2019 at 6:17 PM, BuckGup said:

Yes I need to know exactly how many. The largest value is 1 million.

Oh. Well that's such a small n. You can get away with doing it however you want, including naive trial division.

Here's what I would try in this case, if I can distribute other files with my source (which is frequently not the case with homework):

All non prime numbers n have a prime factor p below the square root of n, and a matching factor on the other side of the square root, 2p. We can use this to build a list of prime numbers below the square root of 1 million.

First, we need a special function, which we'll call edge(root, num), which returns true if and only if root * 2 <= num.

Next, we define our prime counting function pi(a, b). First, we count how many items are in the list less than sqrt(a). If edge(largestItem, a), then we save the number of items * 2 in a variable aPrimes. Else we save (number of items * 2) - 1 in a variable aPrimes. Next, we do the same for sqrt(b), saving the result in bPrimes. Then, we return bPrimes - aPrimes, which is the number of primes between a and b.

ENCRYPTION IS NOT A CRIME

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