If you are referring to the largest "known" prime number, presently it is 2^ 37,156,667-1, an 11,185,272 digit number which was found in Sep 2008 by Hans-Michael Elvenich in Langenfeld near Cologne, Germany.

Any prime number can be expressed in the form 2^p - 1, where p is also a prime, though it is not necessary that every number of the form 2^p - 1, where p is a prime is a prime number.

The numbers of the form 2^p -1 are known as Mersenne prime numbers. With an increasing ability to harness large amounts of computing power it is now possible to calculate larger prime numbers which are Mersenne prime numbers.

The greatest prime number is unbounded. There is no end to numbers - and so to prime numbers.

As the largest number itself is indefinite, the legest prime is also like that. We do not say what is the largest member of a divergent sequence or series. We say a sequence is convergent or covergent to a limit. All divergent sequence has no bound. Whatever great a number you choose, then many greater members we can find.

We say greatest number to a finite set of numbers , or set of prime numbers. We say the greatest number to an infinite sequence of numbers which conveges to a limit. We say the greatest number in a set of numbers which is bouned above. But it is not necessary that even a bouned set or sequence of numbers need not have any greatest number.

Therefore if the question is limited to a list given prime numbers a greatest of the set exists. But of all the set of prime numbers , the greatest number we can not say except unbouded.

A prime number N is defined as one that has only two integral factors, N and 1.

Let use assume Pmax is the largest prime number and all prime numbers from 2 to Pmax, are included in the set {P1, P2, P3 ... Pmax}. It is possible to determine a number greater than Pmax equal to the product P = (P1*P2*P3* ... Pmax) + 1 and this number is a prime as when it is divided by any of the prime numbers smaller than it the remainder is 1. If P is included in the set of prime numbers it would again be possible to determine a larger prime number.

This shows that there are an infinite number of prime numbers and it is meaningless to try to determine the largest prime number.