My initial idea was, having looked at this now, was to vote to close this as not a real question, since to me it seems clearly based on a misunderstanding. But then decided, possibly unwisely, to elaborate on this in more detail as it is getting long and there is also an other issue.

First, let us recall that this is a transcript from some Q&A session at some public lecture.

The question was

Question: What is your thinking about software patents? There is a big discussion going on in Europe right now about whether software should be patentable.

Now, Knuth's reply in full (but emphasis mine).

Knuth: I’m against patents on things that any student should be expected to discover. There have been an awful lot of software patents in the U.S. for ideas that are completely trivial, and that
bothers me a lot. There is an organization that has worked for many years to make patents on all the remaining trivial ideas and then make these available to everyone. The way patenting had
been going was threatening to make the software industry stand still. **Algorithms are inherently mathematical things that should be as unpatentable as the value of π. But for
something nontrivial, something like the interior point method for linear programming,
there’s more justification for somebody getting a right to license
the method for a short time,** instead of keeping it a trade secret. That’s the
whole idea of patents; the word patent means “to make public”.
I was trained in the culture of mathematics, so I’m not used to charging people a penny every time
they use a theorem I proved. But I charge somebody for the time I spend telling them which theorem
to apply. It’s okay to charge for services and customization and improvement, but don’t make
the algorithms themselves proprietary. **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.**

So, first it seems clear to me that the sole idea here is to create a somewhat plausible (*in the context of a public lecture*) scenario where one could make an argument for a patent on a number, or for licensing its use. In the sense that this number could be consider as closer in spirit to some sophisticated numerical-solver than a natural constant (cf the contrasting of interior point method and Pi beforehand).

To further support this idea note that he did not say one computes the GCD, contrary to the impression given in the question, he says (again my emphasis):

by taking the GCD of this constant with an input number, **or by some other funny combination.**

Clearly there is nothing specific referred to, and all this is just there to convince the audience that one could perhaps theoretically envision a situation where there could be an integer such that

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.

He could instead have proclaimed that it is not inconceivable one can with lot of effort find some 1000 binary-digit integer that is particularly well-suited for creating keys for some crypotography-protocol or whatever.

Finally, to just say he suggests the patent system is essentially absurd seems like an overstatement to me. His position on this seems quite a bit more nuanced.

It’s okay to charge for services and customization and improvement, but don’t make the algorithms themselves proprietary.

In that sense, I would even say that for this hyopthetical number he would not find it completely absurd to have some licensing scheme for the number *itself* (yet not for the underlying and surrounding ideas to find it).