32 pixels by 32 pixels in 24-bit color.
Mind bogglingly big. Too big for any real world analogy. We'll have to resort to fantasy to get an idea of just how huge.
Estimates of the total number of sub-atomic particles in the observable universe range from 1065 to 1085. But let's add a few zeroes as a fudge factor to make it a nice round 10100. Now let's say there exist parallel universes that are identical to ours. How many parallel universes? One for each sub-atomic particle in ours: 10100.
That makes the total number of sub-atomic particles in the universe of universes:
10100 * 10100 or 10200
Now suppose that each sub-atomic particle in the universe of universes is a computer
programmed to test the Collatz Conjecture for all possible combinations of bit
patterns in the 2000-bit number. And let's suppose these computers are REALLY fast.
How fast? Each one prints out in one second the same number of solutions as there
are sub-atomic particles in the universe: 10100. (My computer takes 818 seconds to print
one solution. As I said, these computers are REALLY fast. Mind bogglingly fast.)
So, with each of the 10200 computers cranking out 10100 solutions per second, we are generating:
10300 solutions per second
So the question is: How long does it take to solve the 2000-bit number completely?
The answer, of course, is too big to comprehend, so we'll stretch the fantasy as follows:
The universe has been expanding since the moment of the Big Bang, 20 billion years ago. Let's assume that the universe eventually slows its expansion, stops and then collapses (the Big Crunch) at some arbitrary point in the mind bogglingly distant future, say, 10100 seconds (and how long is that?). And further suppose that the universe undergoes successive Big Bang/Big Crunch cycles, as many cycles as there are sub-atomic particles: 10100.
So, all together we have:
10100 Big Bang/Big Crunch cycles; each of which is...
10100 seconds long; during which...
10100 parallel universes; each of which contains...
10100 computers; each of which is printing...
10100 solutions per second.
That gives us a whopping 10500 solutions! Is that enough?
No.
I would need as many copies of the above conglomeration as there sub-atomic particles in the universe just to get to 1% of the solutions to the 2000-bit binary number. I could have said at the outset that
22000 = 10602,
but then you wouldn't have a proper appreciation of how mind bogglingly huge that number is.
By the way, to put the 2000-bit binary number into proper perspective,
consider the icons on your desktop. In true-color video, an icon would be:
Or to put it another way, an icon is a
24,576-bit binary number!
How big is that compared to a 2000-bit number? Mind bogglingly big.
A 24,576 bit number is greater than
107374!
I can't even begin to stretch my fantasy to accommodate a number this large.
Now consider your entire computer screen as a single binary number.
11.5 million bits!!
And how about your entire hard-drive as a single binary number?
80 billion bits!!!!
To find the Collatz solution to a SINGLE worse case 80 billion bit number, it could take
