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# Finding prime numbers

, 13 Sep 2012 CPOL
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This is an alternative for "Finding prime numbers"

## Introduction

This is an even faster and more space efficient variation on the implementation for finding prime numbers using Sieve of Eratosthenes.

## Background

We know that all even numbers greater than 2 are not prime, so remove them from the sieve process a priori.

## Using the code

Like the Clifford Nelson version, I used a simple array of Boolean for numbers. However, numbers are not represented directly. The boolean at index `n` represents the number `2n+1` (for `n > 0`). So, the array can be half the size of the previous version. My timings show this to be about twice as fast.

 Primes up to: Previous Version This Version 2,000,000 7.7 ms 3.14 ms 4,000,000 16.41 ms 7.00 ms

The attached file has the code with both versions, for calculating the timings.

This code below is just the new implementation. Just copy and paste it into a console app:

```class Program
{
private const int repeats = 1000;  // to get more significant timing
private const int rawCount = 2000000;
private const int initStart = 1;
private const int count = 1 + (rawCount - 1) / 2; // 1+ because rawCount-1 just might be prime
private static readonly int countLimit = (((int)Math.Sqrt(rawCount)) - 1) / 2;
private static bool[] _numbers = new bool[count];

static void Main(string[] args)
{
var sw = new System.Diagnostics.Stopwatch();
for (int j = 0; j < repeats; j++)
{
// I excluded initializing the _numbers array from the timing.
for (int i = initStart; i < count; i++)
{
_numbers[i] = true;
}
sw.Start();
Run2();
sw.Stop();
}
Console.WriteLine("Milliseconds/run: {0:F2}", sw.ElapsedMilliseconds/(double)repeats);
// The 1+ of the count is because 2 is assumed to be prime and is not represented in the array.
Console.WriteLine((1 + _numbers.Count(i => i)) + " primes < " + rawCount);
}

private static void Run2()
{
int baseCounter = 0;
int increment;
int index;
while (baseCounter < countLimit)
{
do
{
baseCounter++;
if (baseCounter == count)
return;
} while (!_numbers[baseCounter]);
increment = (baseCounter << 1) + 1;
index = baseCounter + increment;
while (index < count)
{
_numbers[index] = false;
index += increment;
}
}
}
}
```

## Points of Interest

I wondered if it would be possible to assume other small prime factors in the sieve and further reduce the array size? I convinced myself that it is not, since there are prime pairs that differ by two (such as 11 & 13) so any further compression of the sieve array would not be possible (at least for the Sieve of Eratosthenes).

Strangely, both versions exhibit significant slowdown when the size of the sieve array exceeds about 6MB.

This improvement of the sieve is not new! Search Code Project for "Eratosthenes" and you'll find many implementations. Some (probably most) use this type of optimization.

There are other faster methods of finding prime numbers in order, especially for large values, see Sieve of Atkin [^].

## History

9/13/2012 - Initial posting of the Alternative.

## Share

 Software Developer (Senior) Sciex United States
I started programming in Basic on a DECSystem-10 as a Freshman at Caltech in 1974. I quickly transitioned to assembly language, Fortran, and Pascal. As a summer job at JPL, I did analysis of fuel consumption for the Viking Mars Orbiter attitude control system. I also spent a summer doing O/S maintenance at Digital Equipment Corporation.
After graduation, I started developing microprocessor development tools (e.g., cross-compiler, debugger) for Beckman Instruments, a scientific instrument company.
I've worked on custom file-systems, a real-time O/S for Z8000, Expert Systems (SpinPro & PepPro), and internal and external networking support (I was their first webmaster).
I've worked on the DNA analysis system.
I was the console/UI software architect for Ultracentrifuges and protein Capillary Electrophoresis systems.
After 35 years, Danaher having acquired Beckman (now Beckman Coulter), transferred the CE group to become part of Sciex (2014).

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 You are just print the total founded prime number based on input range. Md. Marufuzzaman23-Dec-15 20:03 Md. Marufuzzaman 23-Dec-15 20:03
 just a small trick Siavash _b13-Oct-12 11:37 Siavash _b 13-Oct-12 11:37
 Re: just a small trick Matt T Heffron15-Oct-12 6:44 Matt T Heffron 15-Oct-12 6:44
 Does it not belong to the alternate section of the orinigal version? Ankur\m/13-Sep-12 20:10 Ankur\m/ 13-Sep-12 20:10
 Re: Does it not belong to the alternate section of the original version? Matt T Heffron14-Sep-12 6:49 Matt T Heffron 14-Sep-12 6:49
 Wheel Factorization PIEBALDconsult13-Sep-12 17:22 PIEBALDconsult 13-Sep-12 17:22
 Re: Wheel Factorization Matt T Heffron14-Sep-12 6:47 Matt T Heffron 14-Sep-12 6:47
 You could make it even a bit faster Kenneth Haugland13-Sep-12 14:26 Kenneth Haugland 13-Sep-12 14:26
 I made some small adjustments on Clofford's suggestion and got this: Finding prime numbers[^] Still slower than yours but should indicate that you could improve the code a little.
 I got another ~7% Matt T Heffron14-Sep-12 8:00 Matt T Heffron 14-Sep-12 8:00
 Re: I got another ~7% Kenneth Haugland14-Sep-12 8:28 Kenneth Haugland 14-Sep-12 8:28
 Re: I got another ~7% Matt T Heffron14-Sep-12 11:06 Matt T Heffron 14-Sep-12 11:06
 Re: I got another ~7% Kenneth Haugland14-Sep-12 11:30 Kenneth Haugland 14-Sep-12 11:30
 Re: I got another ~7% Matt T Heffron14-Sep-12 12:04 Matt T Heffron 14-Sep-12 12:04
 Re: I got another ~7% Kenneth Haugland14-Sep-12 12:14 Kenneth Haugland 14-Sep-12 12:14
 Re: I got another ~7% Matt T Heffron14-Sep-12 12:38 Matt T Heffron 14-Sep-12 12:38
 Re: I got another ~7% Kenneth Haugland14-Sep-12 12:49 Kenneth Haugland 14-Sep-12 12:49
 Re: I got another ~7% Matt T Heffron14-Sep-12 12:51 Matt T Heffron 14-Sep-12 12:51
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