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I am trying to find the fastest way to check whether a given number is prime or not (in Java). Below are several primality testing methods I came up with. Is there any better way than the second implementation(isPrime2)?
public class Prime {
public static boolean isPrime1(int n) {
if (n <= 1) {
return false;
}
if (n == 2) {
return true;
}
for (int i = 2; i <= Math.sqrt(n) + 1; i++) {
if (n % i == 0) {
return false;
}
}
return true;
}
public static boolean isPrime2(int n) {
if (n <= 1) {
return false;
}
if (n == 2) {
return true;
}
if (n % 2 == 0) {
return false;
}
for (int i = 3; i <= Math.sqrt(n) + 1; i = i + 2) {
if (n % i == 0) {
return false;
}
}
return true;
}
}
public class PrimeTest {
public PrimeTest() {
}
@Test
public void testIsPrime() throws IllegalArgumentException, IllegalAccessException, InvocationTargetException {
Prime prime = new Prime();
TreeMap<Long, String> methodMap = new TreeMap<Long, String>();
for (Method method : Prime.class.getDeclaredMethods()) {
long startTime = System.currentTimeMillis();
int primeCount = 0;
for (int i = 0; i < 1000000; i++) {
if ((Boolean) method.invoke(prime, i)) {
primeCount++;
}
}
long endTime = System.currentTimeMillis();
Assert.assertEquals(method.getName() + " failed ", 78498, primeCount);
methodMap.put(endTime - startTime, method.getName());
}
for (Entry<Long, String> entry : methodMap.entrySet()) {
System.out.println(entry.getValue() + " " + entry.getKey() + " Milli seconds ");
}
}
}
tested in a Intel Atom @ 1.60GHz, 2GB RAM, 32-bit Operating System
test result:
largest prime number below Long.MAX_VALUE=9223372036854775807 is 9223372036854775783
elapsed time is 171499 milliseconds or 2 minutes and 51 seconds
What you have written is what most common programmers do and which should be sufficient most of the time.
However, if you are after the "best scientific algorithm" there are many variations (with varying levels of certainty) documented http://en.wikipedia.org/wiki/Prime_number.
For example, if you have a 70 digit number JVM's physical limitations can prevent your code from running in which case you can use "Sieves" etc.
Again, like I said if this was a programming question or a general question of usage in software your code should be perfect :)
Algorithm Efficiency : O( n^(1/2)) Algorithm
Note: This sample code below contains count variables and calls to a print function for the purposes of printing the results :
When the prime number 2147483647 is entered, it produces the following output:
Performed 23170 checks, determined 2147483647 is PRIME.
If you are just trying to find if a number is prime or not it's good enough, but if you're trying to find all primes from 0 to n a better option will be the Sieve of Eratosthenes
But it will depend on limitations of java on array sizes etc.
I optimized the trial division here: it returns a Boolean. The methods other than isPrime(n) are also needed.
A fast test due to Jaeschke (1993) is a deterministic version of the Miller-Rabin test, which has no false positives below 4,759,123,141 and hence can be applied to Java
ints.This doesn't work for
longvariables, but a different test does: the BPSW test has no counterexamples up to 2^64. This basically consists of a 2-strong probable prime test like above followed by a strong Lucas test which is a bit more complicated but not fundamentally different.Both of these tests are vastly faster than any kind of trial division.