If you're trying to draw me into a serious discussion on the randomness of
numbers, in general, I have a few comments.
1) The only general purpose way of verifying the actual randomness of a
number is to know by what process it was selected. Your number (for
instance) was likely pulled out of your ass. Unless there is something I
don't know about your ass, it likely wasn't a particularly random number.
2) You can't even begin to speculate about the randomness (or lack
there-of) of a number without knowing the possible range of outputs for
the generating process.
3) A number can be 'random' because it came from a particular process, so
it isn't just the process which is 'random'. Random is an attribute of
numbers, not just processes. In fact, it can even be an attribute of a
_particular_ number.
4) a single sample is a set, it is a set of size
1
All this being said, you can talk about the expected attributes of a
random number: for a good RNG, the each bit is independent of each other
bit and each bit has a 50% chance of being a 1. For instance, your
example (2^3021377 - 1) is a number that requires 3021377 bits to express
(at minimum) to express. If we just take those 3021377 bits, we can do an
analysis of them. (And with a flick of my wrist, we now have an
interesting set of size 3021377) 3021377 bits is more than enough for a
reasonable sample, actually. We would find that the sample contained all
1s. This bit sample would fail any test for statistical randomness that I
can think of.
So, If I were provided only one sample from a RNG that produced 3021377
bit numbers, and this just happened to be the number, I'd say that I'd
reasonably be able to say that the RNG was poor.
---snip---
You'll note that I did a bitwise analysis of the sample. Though any
_particular_ number in my RNG has a 1/2^3021377 chance in being selected,
the bits that make up that number have different characteristics.
Like you can still expect the sample to generally display certain
statistical properties. Because the sample does not does not display
these statistical properties, you can draw one of two conclusions: 1) Wow,
coincidences _do_ happen, I guess I should choose again or 2) OK,
coincidence is one thing, but this particular occurrence is has about the
same probability as all the molecules of air just happening to be in the
other side of the room, causing spontaneous human decompression (ie: it
could happen, but in practice it doesn't). :-) It doesn't help that 'all
ones' is a common RNG failure.
---snip---
It is the difference between possibility and likelihood. Sure, there is a possibility that the RNG just happens to spit out 2^3021377 - 1. In fact, assuming that the RNG generates 3021377 bit numbers, the possibility is 1/2^3021377. Also true, is that any other _particular_ number has the same likelihood to be picked as 2^3021377 - 1 (assuming it is a good RNG). But if you just change your view, you can comment on the number more rigorously. You can break up the number into lots of uniformly sized sections and analyze them. Because of the nature of randomness, you can expect any arbitrarily sized sub-section of the random number to display the same properties as the whole. (this is trivially provable, and is left as an exercise for the reader. LOL. If anyone is interested in why this is, I'll address it separately)
I did a quick wave of my hands at the 1 bit sized chunks, but randomness testing is by no means limited to this. At my work, we do a series of statistical tests to attempt to tell if the PRNG in use is terribly, terribly flawed. (often it is... ). We test for a series of bit-based properties, and a nibble-based property. This is a not a particularly thorough test, but it catches lots of people. If I were interested in going beyond this, I'd do similar tests with larger chunks (at least 8 bit, and maybe 16 bit chunks). If I _really_ cared, I'd also do an additional series of tests (which, once again, if you're interested, I can go into...)
OK, so what I've been trying to get to is this: just because each number is equally likely to occur, doesn't mean that every attribute of the number is also equally as likely. For instance, you can say:
attribute: number of 1s in a 4 bit number.
If you examine the probability distribution for this attribute, you'll find that there is exactly 1 case in the 2^4 sized space that has the attribute of all '1's, whereas there are 6 cases that have 2 '1's. So, obviously for a 4 bit number, you are more likely to get a number with 2 '1's than you are to get a number with 4 '1's. But not by a _huge_ amount: P(2 '1's) = 6/16 = .375, whereas P(0 '1's) = 1/16 = .0625.
Note for further calculations: The generic way of computing the count of numbers that contain a certain count of '1's is this: for n bit numbers, the count of numbers containing m '1' bits is Combo(n, m) (that is n choose m).
Now lets look at the 3021377 bit number. There are Combo(3021377, 1510688) (about 10^909521 or 2^3021363) numbers where the count of '1's is 1510688 (and the same count of numbers where the count of '1's is 1510689). There is 1 number with 3021377 '1's. So, P(1510688 '1's) = 1/2^14 = 6.103E-5, whereas P(3021377, 3021377) = 1 / 2^3021377 = 1E-909525. So, the all '1's set is 10^909519 time less likely than the roughly half 1s set.
That's a big number. So, though you can say that every output number is equally likely, you can also say that you can expect the output number to be comprised of roughly half '1' bits. It sounds strange, but it's true.