Mersenne numbers, when they are primes, are routinely used to feed MonteCarlo random number generation. They have the form:

.

Since not all of them are prime, we can use prime factorization of a sequence of big Mersenne numbers to benchmark our CPU.

The ingredients are fairly minimalistic:

all we need is a Linux bash shell (with the wonderful bench calculator bc), the java compiler (Linux: javac) and the java virtual machine (Linux: java). To factorize we could use for example the Pollard Rho method in this implementation.

import java.math.BigInteger;
import java.security.SecureRandom;
class PollardRho {
private final static BigInteger ZERO = new BigInteger("0");
private final static BigInteger ONE = new BigInteger("1");
private final static BigInteger TWO = new BigInteger("2");
private final static SecureRandom random = new SecureRandom();
public static BigInteger rho(BigInteger N) {
BigInteger divisor;
BigInteger c = new BigInteger(N.bitLength(), random);
BigInteger x = new BigInteger(N.bitLength(), random);
BigInteger xx = x;
// check divisibility by 2
if (N.mod(TWO).compareTo(ZERO) == 0) return TWO;
do {
x = x.multiply(x).mod(N).add(c).mod(N);
xx = xx.multiply(xx).mod(N).add(c).mod(N);
xx = xx.multiply(xx).mod(N).add(c).mod(N);
divisor = x.subtract(xx).gcd(N);
} while((divisor.compareTo(ONE)) == 0);
return divisor;
}
public static void factor(BigInteger N) {
if (N.compareTo(ONE) == 0) return;
if (N.isProbablePrime(20))
{ System.out.println(N); return; }
BigInteger divisor = rho(N);
factor(divisor);
factor(N.divide(divisor));
}
public static void main(String[] args) {
BigInteger N = new BigInteger(args[0]);
factor(N);
}
}

If processor time has to be accurately recorded, just add the “time” command in front of the java runtime, like this:

for n in `seq 101 2 201`; \
do echo "[Factoring Mersenne `echo 2^$n-1`]" ; \
time java PollardRho `echo 2^$n -1 |bc` ; \
done

The code snippet above loops all the odd numbers , uses them as Mersenne exponent as and prints prime factors. Check for example that for the Mersenne number is infact a Mersenne prime.

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