TITLE: shallow question: Why a 300 digit number is associated with "any NP-hard problem"?
QUESTION [8 upvotes]: I was reading this article (http://www.ams.org/notices/200203/fea-knuth.pdf) the other day and noticed Donald Knuth said something nontrivial: Theoretically we can compute a very large number of magnititude $10^{300}$ (or possibly even larger) and use it via taking greatest common divisor to solve any NP hard problem. He used this example to suggest the patent system is essentially absurd. 
Here is the copy of the original paragraph:
"There’s an interesting issue, though. Could you
possibly have a patent on a positive integer? It is
not inconceivable that if we took a million of the
greatest supercomputers today and set them going,
they could compute a certain 300-digit constant
that would solve any NP-hard problem by taking
the GCD of this constant with an input number, or
by some other funny combination. This integer
would require massive amounts of computation
time to find, and if you knew that integer, then you
could do all kinds of useful things. Now, is that
integer really discovered by man? Or is it something
that is God given? When we start thinking of complexity issues, we have to change our viewpoint as
to what is in nature and what is invented."
My shallow question(must have already been asked by someone) is: Why finding such a large integer would suffice to solve any $NP$-hard problem? I asked a friend majoring in algorithm design in computer science, and he could not tell me the answer as well. While he explained to me that a $NP$ hard problem can be reduced to finding the GCD, neither he nor I can understand why finding one number is suffice for all applications. I am not sure if this is more appropriate for mathoverflow or stackoverflow.

REPLY [2 votes]: No matter how you formalize the question, if we believe that this world is not a computer simulation with a program of decent length with all randomness introduced by some simple standard pseudorandom generator but a truly random chain of events, then it looks like a rare moment of true unforgivable ignorance of Donald Knuth. He had no excuse for not knowing http://en.wikipedia.org/wiki/Kolmogorov_complexity. It is actually much easier for me to believe that gods had a direct communication with him and showed him a part of the program somehow than to believe that he didn't know what he was talking about to the extent described. Well, perhaps this was the case and we are just funny configurations in some game with simple rules (it was Knuth who designed "life", wasn't it?) but I prefer to act under the alternative assumptions, at least for the next few years :).