The Java BigInteger class has a method .isProbablePrime(int certainty) which claims to return true if a given number is likely prime up to the requested level of certainty.

I tested Elon's prime up to a certainty of 10,240, and it returned true.

This means that there is less than one chance in 10^3000 that the number is composite.

From the JavaDoc:

Returns true if this BigInteger is probably prime, false if it's definitely composite. If certainty is ≤ 0, true is returned.

Params:

certainty – a measure of the uncertainty that the caller is willing to tolerate: if the call returns true the probability that this BigInteger is prime exceeds (1 - (1/2) ^ certainty). The execution time of this method is proportional to the value of this parameter.

Returns: true if this BigInteger is probably prime, false if it's definitely composite.

Here's the Java code (String taken from above C++ code. Thanks!)

package misc.primes;

import java.math.*;

public class XPrime { static final String xPrimeString = "111111111111111111111111111111111111111111111111111111111111" + //line 01 "111111111111111111111111111111111111111111111111111111111111" + //line 02 "111111111111111111111111111111111111111111111111111111111111" + //line 03 "111111111111111111111111111111111111111111111111111111111111" + //line 04 "111111111111111111111111111111111111111111111111111111111111" + //line 05 "111111111111111111111111111111111111111111111111111111111111" + //line 06 "111111111111188888888888111111111111111111888811111111111111" + //line 07 "111111111111118888888888881111111111111188888111111111111111" + //line 08 "111111111111111888811118888111111111118888811111111111111111" + //line 09 "111111111111111118888111888811111111188881111111111111117111" + //line 10 // originally ends 7111 "111111111111111111888811188888111118888811111111111111111111" + //line 11 "111111111111111111188888111888811888881111111111111111111111" + //line 12 "111111111111111111111888811188888888111111111111111111111111" + //line 13 "111111111111111111111188881118888811111111111111111111111111" + //line 14 "111111111111111111111118888811188881111111111111111111111111" + //line 15 "111111111111111111111111188881118888811111111111111111111111" + //line 16 "111111111111111111111111188888811188881111111111111111111111" + //line 17 "111111111111111111111118888888881118888111111111111111111111" + //line 18 "111111111111111111111888881118888111888881111111111111111111" + //line 19 "111111111111111111118888111111888881118888111111111111111111" + //line 20 "111111111111111111888881111111118888111888811111111111111111" + //line 21 "111111111111111188888111111111111888811118888111111111111111" + //line 22 "111111111111118888811111111111111188888888888811111111111111" + //line 23 "111111111111188881111111111111111111888888888881111111111111" + //line 24 "111111111111111111111111111111111111111111111111111111111111" + //line 25 "111111111111111111111111111111111111111111111111111111111111" + //line 26 "111111111111111111111111111111111111111111111111111111111111" + //line 27 "111111111111111111111111111111111111111111111111111111111111" + //line 28 "111111111111111111111111111111111111111111111111111111111111" + //line 29 "111111111111111111111111111111111111111111111111111111111111";

static final BigInteger xPrime = new BigInteger(xPrimeString);

public static void main(final String[] args)
{
	final boolean isPrime = xPrime.isProbablePrime(10240);

	System.out.println("Is Prime: " + isPrime);
}

}