I don't think it's impossible to prove, I think we just aren't yet good at proving it. One reason I believe P!=NP is I can imagine what a proof of P=NP would l

I don't think it's impossible to prove, I think we just aren't yet good at proving it. One reason I believe P!=NP is I can imagine what a proof of P=NP would look like -- just write a C program that solves SAT or Sudoku or something, in polynomial time, and prove it's correctness and execution time. There are tens of thousands of papers which do that, for various problems. I can't really imagine what a proof of P!=NP would look like, proving something doesn't exist is extremely hard, particularly because for any particular instance of an NP problem, there is a problem which solves it "by luck".