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Old 2003-11-24, 16:49   #1
danjmi
 
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Default Largest number in the real universe

In explaining GIMPS to friends, I tell them that the numbers we're dealing with vastly exceed anything that could possibly have meaning in the real universe.
My thinking is that the biggest meaningful number would be the dimensions of space-time to the distance light could have travelled since the Big Bang, measured in Planck units. Even multiplying by factors for different states of the elementary cells, or the hidden dimensions hypothesized by cosmologists, it seems to come to only 2 to a power of a few thousands.
Has anyone made calculations along these lines?
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Old 2003-11-24, 17:13   #2
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Here is Cheesehead's answer to your question.

It appeared in the Math forum

-------8<--------8<------------8<-------------
A while ago, while writing a posting illustrate the magnitudes of the numbers we work with, I looked up current estimates of the numbers of particles in the known universe, size of the known universe, and related stuff.

Without giving links or specific citations, here is a rough calculation to demonstrate the impossibility of trial-factoring to the square root of the size of Mersenne number with which GIMPS is now working:

The known universe could hold (far) less than 10^200 neutrons (the most compact elementary particle) if it were packed full, with no empty space. (In reality, the universe is more than 99.99% empty.)

The "Planck time", which is smaller than any time in which any conceivable useful computation could take place, is greater than 10^-44 second. Let's make that 10^-100 second, just so there's no quibbling. So, no computation could be performed in less than 10^-100 second.

Suppose the known universe were packed full of neutrons, and suppose each neutron were a computer capable of performing one trial-factoring division in 10^-100 second. So, altogether the universe could perform 10^300 trial-factoring divisions per second.

Now, 10^300 is less than 2^1200, so all the computers in the entire known universe can perform no more than 2^1200 trial-factoring divisions per second.

The estimated age of the known universe is 13 billion years. There are about 31 million seconds in a year. So the universe is about 400 million billion seconds old. That's 4 * 10^17 seconds, which is (much) less than 2^100 seconds.

So all the computers in the universe, operating at the fastest possible speed for longer than the age of the universe, could perform no more than 2^1300 trial-factoring divisions.

What is the size of a number than has 2^1300 primes below its square root?

Let N be the number, and Q be its square root.

Pi(Q) = 2^1300

Pi(Q) = approx Q / ln Q

{Edit: Here I'm making a WAG ("educated guess":) that Q is about 2^1400 --} The natural log of 2^1400 would be less than 1400 (because e^1400 > 2^1400).

So if Q were 2^1400, then Pi(Q) would be greater than (2^1400)/1400 > 2^1380.

So, we know Q &lt; 2^1400, and so N &lt; 2^2800.

In other words, if the entire known universe were packed full of neutrons and each neutron were a computer operating at maximum possible speed, and all the little computers ran for the entire age (so far) of the universe, it could completely trial-factor a number no larger than 2^2800.

WAIT! We left out the optimizations -- like each factor has to be 2kp+1 and +-1 mod 8, and so on.

Suppose our optimizations allow us to skip 999,999,999,999 out of every 1,000,000,000,000 primes below the square root of the number we're trying to factor. That means we can TF a trillion (~2^40) times as many potential factors. So we want Pi(Q) = 2^1300 * 2^40 = 2^1340 instead of 2^1300.

Hmmm ... looking back, we find that "So if Q were 2^1400, then Pi(Q) would be greater than (2^1400)/1400 > 2^1380" still is valid. We don't have to change our previous answer -- The entire universe could TF a number no larger than 2^2800. (Using our current trial-factoring methods and optimizations, that is.)

{EDIT: Let me restate that conclusion so that it is clearer when quoted out of context -- Even if the entire known universe were one solid computer operating at maximum speed for the entire time since the Big Bang, it could not yet have trial-factored a number larger than 2^2800 all the way to its square root.}

Let's see ... how big are the numbers GIMPS is working on now? Current PrimeNet trial-factoring assignments are greater than M21000000 = 2^21000000 - 1 which is far, far larger than 2^2800.
-------8<--------8<------------8<-------------

Hope this helps...

Luigi
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Old 2003-11-24, 17:59   #3
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First: I have compiled my answer before reading ET_s interesting reply. I will post it nonetheless unchanged

10^18 seconds universe age
10^44 planck time quants/second

(10^62)^3 (hey, my universe is a cube) cells

we are at 10^184 cells here

Those 10^184 cells exist in a different state at each of the 10^62 time units in one of the allowed ??? (i also 'estimate' 10^62 for whatever reason here) energy states. Roughly 10^300 here... So far so good and compatible with your estimate

But how about allowing spontanus quantum teleportations "between" two time units? Would that not lead to (10^184)! possible permutations of the cell contents? And that would easily

This is really only speculation and maybe plain nonsense - but fun ;)

Remark and shameless math-speculation:

Those 10^300something are in the region, where the first crossing of pi(n)-li(n)=0 has to occur. (http://mathworld.wolfram.com/PrimeNumberTheorem.html). Maybe that is also the region, where the primes are no longer distributed (normalized) 'random'? The 10^300something number also looks like the maximum abount of information (bits) that make sense in a pysical sense. So if the null-values of the Zeta-function really are linked to quantum dynamical systems (see the Montgomery-Odlyzko Law) _and_ you take our universe as a whole as that quantum dynamical system, the region 10^300 may well be the region, where the "random" distribution of primes is no longer _required_ to support the nature of our universe and the RH turns out to be false :)

Ok - i'll take my pills again and be quite *g*

P.S.: Check http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm for _really_ big numbers

Tau
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Old 2003-11-24, 18:42   #4
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(much snippage)

Quote:
Originally posted by ET_

Hope this helps...

Luigi
Vary cool. I'd also note that there's a fundamental limit on the number of neutrons you can "pack" - the Schwartzchild limit - beyond that you get a black hole. Without looking it up, for packed neutrons that's less than 30 miles in diameter (and I believe it's a LOT less).

There are numbers which can be described that are far bigger than those GIMPS is working with - look up Knuth arrows, Conway arrows, and the Ackermann (sp?) function - but those numbers also have little to no physical meaning.
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Old 2003-11-24, 21:25   #5
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Hey, waitaminnit! :-P

I only posted Cheesehead message, he's the guru, not me :-)

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Old 2003-11-24, 23:50   #6
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If, and insofar as, it is in my "authority" to do so, I hereby appoint Luigi to be acting guru of this thread, whether he wants to be or not. :-)

Luigi, every guru has to start somewhere. (Want me to send you a cheesehat catalog?)

Last fiddled with by cheesehead on 2003-11-24 at 23:57
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Old 2003-11-25, 07:03   #7
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A lower upper-bound could probably be obtained by considering themodynamic arguments.
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Old 2004-08-16, 22:10   #8
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i believe there is no largest number they just go on and on.
 
Old 2004-08-17, 02:04   #9
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That's an interesting question...Is there a finite upper bound to the number of things that have ever existed? The number of particles that ever existed, plus all their quantum states including superposition of states, plus each change of state, plus each fundamental unit of time or distance if they exist, etc.
 
Old 2004-08-17, 12:06   #10
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Quote:
Originally Posted by Unregistered
That's an interesting question...Is there a finite upper bound to the number of things that have ever existed? The number of particles that ever existed, plus all their quantum states including superposition of states, plus each change of state, plus each fundamental unit of time or distance if they exist, etc.
รน

I think this is the modern question to the ancient dicotomy between potential and actual Infinite.

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Old 2004-08-21, 19:21   #11
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Quote:
Originally Posted by Unregistered
Is there a finite upper bound to the number of things that have ever existed?
I would argue that it boils down to the question whether the universe is finite or infinite.. If it was finite, the number of well-defined entities in it should be countable; It is not known if the Universe is infinite in time/extension.. But the "granularity" could be a severe limit to counting.. Measuring quantum states is impossible AFAIK, so you could at best form an average of "things" that probably ever existed.
 
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