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    <p>Here's another way:</p> <pre><code>boolean isPrime(long n) { if(n &lt; 2) return false; if(n == 2 || n == 3) return true; if(n%2 == 0 || n%3 == 0) return false; long sqrtN = (long)Math.sqrt(n)+1; for(long i = 6L; i &lt;= sqrtN; i += 6) { if(n%(i-1) == 0 || n%(i+1) == 0) return false; } return true; } </code></pre> <p>and <a href="http://java.sun.com/javase/6/docs/api/java/math/BigInteger.html#isProbablePrime%28int%29" rel="noreferrer"><code>BigInteger's isProbablePrime(...)</code></a> is valid for all 32 bit <code>int</code>'s.</p> <p><strong>EDIT</strong></p> <p>Note that <code>isProbablePrime(certainty)</code> does not always produce the correct answer. When the certainty is on the low side, it produces false positives, as @dimo414 mentioned in the comments.</p> <p>Unfortunately, I could not find the source that claimed <code>isProbablePrime(certainty)</code> is valid for all (32-bit) <code>int</code>'s (given enough certainty!).</p> <p>So I performed a couple of tests. I created a <code>BitSet</code> of size <code>Integer.MAX_VALUE/2</code> representing all uneven numbers and used a prime sieve to find all primes in the range <code>1..Integer.MAX_VALUE</code>. I then looped from <code>i=1..Integer.MAX_VALUE</code> to test that every <code>new BigInteger(String.valueOf(i)).isProbablePrime(certainty) == isPrime(i)</code>.</p> <p>For certainty 5 and 10, <code>isProbablePrime(...)</code> produced false positives along the line. But with <code>isProbablePrime(15)</code>, no test failed.</p> <p>Here's my test rig:</p> <pre><code>import java.math.BigInteger; import java.util.BitSet; public class Main { static BitSet primes; static boolean isPrime(int p) { return p &gt; 0 &amp;&amp; (p == 2 || (p%2 != 0 &amp;&amp; primes.get(p/2))); } static void generatePrimesUpTo(int n) { primes = new BitSet(n/2); for(int i = 0; i &lt; primes.size(); i++) { primes.set(i, true); } primes.set(0, false); int stop = (int)Math.sqrt(n) + 1; int percentageDone = 0, previousPercentageDone = 0; System.out.println("generating primes..."); long start = System.currentTimeMillis(); for(int i = 0; i &lt;= stop; i++) { previousPercentageDone = percentageDone; percentageDone = (int)((i + 1.0) / (stop / 100.0)); if(percentageDone &lt;= 100 &amp;&amp; percentageDone != previousPercentageDone) { System.out.println(percentageDone + "%"); } if(primes.get(i)) { int number = (i * 2) + 1; for(int p = number * 2; p &lt; n; p += number) { if(p &lt; 0) break; // overflow if(p%2 == 0) continue; primes.set(p/2, false); } } } long elapsed = System.currentTimeMillis() - start; System.out.println("finished generating primes ~" + (elapsed/1000) + " seconds"); } private static void test(final int certainty, final int n) { int percentageDone = 0, previousPercentageDone = 0; long start = System.currentTimeMillis(); System.out.println("testing isProbablePrime(" + certainty + ") from 1 to " + n); for(int i = 1; i &lt; n; i++) { previousPercentageDone = percentageDone; percentageDone = (int)((i + 1.0) / (n / 100.0)); if(percentageDone &lt;= 100 &amp;&amp; percentageDone != previousPercentageDone) { System.out.println(percentageDone + "%"); } BigInteger bigInt = new BigInteger(String.valueOf(i)); boolean bigIntSays = bigInt.isProbablePrime(certainty); if(isPrime(i) != bigIntSays) { System.out.println("ERROR: isProbablePrime(" + certainty + ") returns " + bigIntSays + " for i=" + i + " while it " + (isPrime(i) ? "is" : "isn't" ) + " a prime"); return; } } long elapsed = System.currentTimeMillis() - start; System.out.println("finished testing in ~" + ((elapsed/1000)/60) + " minutes, no false positive or false negative found for isProbablePrime(" + certainty + ")"); } public static void main(String[] args) { int certainty = Integer.parseInt(args[0]); int n = Integer.MAX_VALUE; generatePrimesUpTo(n); test(certainty, n); } } </code></pre> <p>which I ran by doing:</p> <pre><code>java -Xmx1024m -cp . Main 15 </code></pre> <p>The generating of the primes took ~30 sec on my machine. And the actual test of all <code>i</code> in <code>1..Integer.MAX_VALUE</code> took around 2 hours and 15 minutes.</p>
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    2. CO`(long)Math.sqrt(n)` should've been `(long)Math.sqrt(n)+1`
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    3. COisPrime3 2213 Milli seconds isPrime2 3039 Milli seconds isPrime1 6030 Milli seconds you beat me
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