It's very cool finding the largest known prime number, but it's definitely not the largest prime number that exists, as in fact, there are INFINITELY many prime numbers...
To prove this, let's say that there was a finite number of prime numbers. Then there would exist a prime number that is the largest. Call it P.
Create a number by multiplying all the prime numbers up to P, so
N = 2 x 3 x 5 x 7 x 11 x ... x P
If we add 1 to this number, we get
N + 1 = 2 x 3 x 5 x 7 x 11 x ... x P + 1
Now if we were to divide N + 1 by any of the prime numbers up to P, there will always be a remainder of 1.
So that means as N + 1 does not have any prime factors up to P, then either there is a prime number larger than P which divides N + 1, or else N + 1 itself is a prime number. Either way, this contradicts our statement that P is the largest prime number, and thus our original statement that there are a finite number of prime numbers must be false.
Thus there are an infinite number of prime numbers. Q.E.D.