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    <p>Many thanks to all who gave helpful answers. Here are my implementations of a few different methods of finding the first <em>n</em> primes in C#. The first two methods are pretty much what was posted here. (The posters names are next to the title.) I plan on doing the sieve of Atkin sometime, although I suspect it won't be quite as simple as the methods here currently. If anybody can see any way of improving any of these methods I'd love to know :-)</p> <p><strong>Standard Method</strong> (<a href="https://stackoverflow.com/questions/1042902/most-elegant-way-to-generate-prime-numbers/1043007#1043007">Peter Smit</a>, <a href="https://stackoverflow.com/questions/1042902/most-elegant-way-to-generate-prime-numbers/1043002#1043002">jmservera</a>, <a href="https://stackoverflow.com/questions/1042902/most-elegant-way-to-generate-prime-numbers/1042969#1042969">Rekreativc</a>)</p> <p>The first prime number is 2. Add this to a list of primes. The next prime is the next number that is not evenly divisible by any number on this list.</p> <pre><code>public static List&lt;int&gt; GeneratePrimesNaive(int n) { List&lt;int&gt; primes = new List&lt;int&gt;(); primes.Add(2); int nextPrime = 3; while (primes.Count &lt; n) { int sqrt = (int)Math.Sqrt(nextPrime); bool isPrime = true; for (int i = 0; (int)primes[i] &lt;= sqrt; i++) { if (nextPrime % primes[i] == 0) { isPrime = false; break; } } if (isPrime) { primes.Add(nextPrime); } nextPrime += 2; } return primes; } </code></pre> <p>This has been optimised by only testing for divisibility up to the square root of the number being tested; and by only testing odd numbers. This can be further optimised by testing only numbers of the form <code>6k+[1, 5]</code>, or <code>30k+[1, 7, 11, 13, 17, 19, 23, 29]</code> or <a href="http://en.wikipedia.org/wiki/Prime_number#Examples_and_first_properties" rel="nofollow noreferrer">so on</a>.</p> <p><strong>Sieve of Eratosthenes</strong> (<a href="https://stackoverflow.com/questions/1042902/most-elegant-way-to-generate-prime-numbers/1043247#1043247">starblue</a>)</p> <p><a href="http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" rel="nofollow noreferrer">This finds all the primes to <em>k</em></a>. To make a list of the first <em>n</em> primes, we first need to approximate value of the <em>n</em>th prime. The following method, <a href="https://stackoverflow.com/questions/1042717/is-there-a-way-to-find-the-approximate-value-of-the-nth-prime/1069023#1069023">as described here</a>, does this.</p> <pre><code>public static int ApproximateNthPrime(int nn) { double n = (double)nn; double p; if (nn &gt;= 7022) { p = n * Math.Log(n) + n * (Math.Log(Math.Log(n)) - 0.9385); } else if (nn &gt;= 6) { p = n * Math.Log(n) + n * Math.Log(Math.Log(n)); } else if (nn &gt; 0) { p = new int[] { 2, 3, 5, 7, 11 }[nn - 1]; } else { p = 0; } return (int)p; } // Find all primes up to and including the limit public static BitArray SieveOfEratosthenes(int limit) { BitArray bits = new BitArray(limit + 1, true); bits[0] = false; bits[1] = false; for (int i = 0; i * i &lt;= limit; i++) { if (bits[i]) { for (int j = i * i; j &lt;= limit; j += i) { bits[j] = false; } } } return bits; } public static List&lt;int&gt; GeneratePrimesSieveOfEratosthenes(int n) { int limit = ApproximateNthPrime(n); BitArray bits = SieveOfEratosthenes(limit); List&lt;int&gt; primes = new List&lt;int&gt;(); for (int i = 0, found = 0; i &lt; limit &amp;&amp; found &lt; n; i++) { if (bits[i]) { primes.Add(i); found++; } } return primes; } </code></pre> <p><strong>Sieve of Sundaram</strong></p> <p>I only discovered <a href="http://en.wikipedia.org/wiki/Sieve_of_Sundaram" rel="nofollow noreferrer">this sieve</a> recently, but it can be implemented quite simply. My implementation isn't as fast as the sieve of Eratosthenes, but it is significantly faster than the naive method.</p> <pre><code>public static BitArray SieveOfSundaram(int limit) { limit /= 2; BitArray bits = new BitArray(limit + 1, true); for (int i = 1; 3 * i + 1 &lt; limit; i++) { for (int j = 1; i + j + 2 * i * j &lt;= limit; j++) { bits[i + j + 2 * i * j] = false; } } return bits; } public static List&lt;int&gt; GeneratePrimesSieveOfSundaram(int n) { int limit = ApproximateNthPrime(n); BitArray bits = SieveOfSundaram(limit); List&lt;int&gt; primes = new List&lt;int&gt;(); primes.Add(2); for (int i = 1, found = 1; 2 * i + 1 &lt;= limit &amp;&amp; found &lt; n; i++) { if (bits[i]) { primes.Add(2 * i + 1); found++; } } return primes; } </code></pre>
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