Computer scientist here. I agree with some of the sentiments of the other replies; indeed, most of your readership will probably glance over any statements about P vs. NP and take your word for it, at least as the problem pertains to your story. And generally speaking, I am more on the side that favors storytelling over fact (when that is necessary.) However, I'll try to give you my impression of your ideas as someone who studied the subject, because I appreciate attention to detail, and because I believe that knowing what you're writing about, even if you don't use that knowledge explicitly, can give you a better feel for the underlying ideas and result in a better story.
He explains the P versus NP problem such that computers can solve some problems (P) and people are needed to solve others (NP).
Honestly, this is the first time I'm encountering such a definition of P and NP. Now, obviously, this is not a rigorous definition and it isn't intended to be, but it also gives the wrong impression that humans are somehow related to the mathematical objects P and NP. To be clear, humans play no role in complexity theory. However, I can kind-of see where you want to go with this, and it might work. In a way. I'll talk about exactly how in a bit.
Is there anything I'm missing in the complexities of P and NP?
I could go into the precise definitions in layman's terms, but that'd be long and probably unhelpful. So I'll try and be concise: P is the set of problems that can be solved in a reasonable time by computers, while NP is the set of problems that can be verified in a reasonable time by computers. Let's for now think about a problem as a question that one needs to answer. If it's in P, that means that there is a fast algorithm that can answer that question. If it's in NP, that means that there is a fast algorithm which, given an answer, can tell you if that answer is correct. Again, questions in P can be answered easily, but for questions in NP, you are only guaranteed that any answer
can be checked easily. But where do you get the answer from? That might take a long long time to compute.
That's where you might want to insert "human intuition." Maybe there's something special about our brains, that allows us to arrive at solutions without fully understanding how we got there? That allows us to pull answers from the ether, and then we only need to check them? Personally, I don't buy it, but that besides the point.
Think about Sudoku, for example. It might take a long time to solve, so it's not necessarily in P. But if someone presents you with a solution, it's very easy to check that the solution is correct. (In fact, you guessed it,
Sudoku is NP-complete.) But maybe, somehow, human intuition can help us solve Sudoku efficiently if combined with a computer?
If brain-uploading is possible in this story, does the rest of the idea check out?
Personally, I think the main attraction of mind-uploading is not necessarily proving P=NP (it has nothing to do with that), but with being able simulate human brains (a form of AI) much faster than you an I can think. Imagine the progress that can be made by AI's that think 1000 times faster than us. This is not about breaking some complexity barrier, complexity hasn't got much to do with it. This is about speeding up intellectual progress. In a year you can get the same progress it would take humanity 1000 years to achieve (so, yeah, maybe then the AI's would have already figured out P vs. NP?)
So, forgive the detours, and let me lay out exactly in what way I think this might work. I see basically two ways forward:
(1) As I said, uploading human minds and speeding them up may yield a proof for P=NP, simply by having human minds working on it for long. That proof may very well include a way to solve
any problem quickly, which is basically what you want from proving P=NP, I guess.
(2) One may claim that the solution for P vs. NP was in the human mind all along. How? Well, the way to show P=NP is to show that any problem that can be checked quickly, can also be solved quickly. But there's an easier way.
NP-completeness tells us that really, you only need to show that
one problem can be solved quickly, and maybe the solution to such a problem can be achieved by running the human mind more efficiently. Such problems are called NP-complete problems, and there are several of them known (like Sudoku), but obviously none of them are known to be in P. Your mathematician may have found a way to solve an NP-complete problem by combining brains and machines, which would indeed prove P=NP. (That could
never happen in reality, but who cares.) By the way, the concept of NP-completeness is a pretty incredible thing, if you care to get into it, but it might be too technical for the story, idk.
Oh, another thing you might want to consider. If P=NP, then something called the Hierarchy Collapse will occur. (Dramatic, isn't it?) This is basically the collapse of the
Polynomial Hierarchy, which is something even more profound than P=NP, albeit much less well-known to the public. It would mean that pretty much any question that you can ask can be answered quickly (well, kinda, but again, who cares.) Honestly, just the name Hierarchy Collapse sounds cool.
Anyway, I wrote a lot, and I hope I helped. And again, like the others said, put your story and readers first, fact second. After all, P=NP is a pretty radical statement to make in
our world, but it's completely acceptable in yours.