Well, it is what it is (the first prime number after a googolplex). What would you prefer as a description of it? A listing of its digits? That would take rather a long time to read out. I can start you off, though. Its first few digits are 100000....
(It's easy enough to write a program that will eventually, slowly, find and stream to the display its digits or any other basic information about it you are interested in, of course. It's just infeasibly slow to use such a thing at the moment.)
The current largest "known" prime, for some sense of what it is to indicate a specific number and know it to be prime, has about 20 million digits (and a very specific convenient form). The prime you are looking for will have about a googol digits instead. So, we're far off, at the moment, from being able to figure out the sort of information about it you're presumably interested in.
Well, we could represent it by googolplex + x. I believe x cannot be 1 for it to be prime due to number theoretical reasonings that I forgot in the meantime.
So, we're far off, at the moment, from being able to figure out the sort of information about it you're presumably interested in.
I'm not "presumably interested". I'm genuinely interested and curious about it. I'm just hesitant to invest a lot of time in it because of the unlikelihood of success.
Just to clarify, when I said "the sort of information about it you're presumably interested in", I wasn't at all doubting your genuine interest and curiosity, but rather, whether I was correct in my assumptions as to the sort of information about the relevant number that said interest and curiosity would be directed towards.
Mostly, it's just a hedging phrase that I felt the need to write which carries no great meaning. (And, in some sense, this whole post is similar...)
Imagine primality testing was O(p) where p is the number of bits in the number you want to test. It is, in fact, way slower but let's give ourselves a huge advantage. Testing any googolplex-sized number would still be impossible. This seems to answer your question fairly simply and conclusively.
I agree with you, there is no way we could compute it with our computers. But there is still number theory with which you can analyze incomprehensibly large numbers and make statements about them. Maybe you could say that the next 10 numbers after googolplex are not prime. That would already be a first step towards an answer. You can't just categorically say it's impossible to find out. It is just that our understanding of mathematics is far from being able to answer this question. Anyway, it's more of a playful question I'm asking just for fun. Whether or not we find an answer is not of importance to me.
It's a perfectly fun and good question, I just don't think the answer is that hard. Short of a truly titanic mathematical breakthrough, it's not possible to find out. Beside this being one of the oldest, most worked class of problems in mathematics, the people doing these large and systematic searches are not ignorant of number theory:
While you can easily tell some numbers around googolplex are not prime (googolplex + 1 and googolplex + 2 are obviously not) this doesn't tell you anything useful about where the next prime might be. Testing a single such gigantic number for primality is infeasible and there aren't exactly a lot of prime numbers out there - on average, one every couple of googols as you can see from:
(It's easy enough to write a program that will eventually, slowly, find and stream to the display its digits or any other basic information about it you are interested in, of course. It's just infeasibly slow to use such a thing at the moment.)
The current largest "known" prime, for some sense of what it is to indicate a specific number and know it to be prime, has about 20 million digits (and a very specific convenient form). The prime you are looking for will have about a googol digits instead. So, we're far off, at the moment, from being able to figure out the sort of information about it you're presumably interested in.