20031124, 16:49  #1 
Nov 2003
2^{2} Posts 
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 spacetime 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? danjmi 
20031124, 17:13  #2 
Banned
"Luigi"
Aug 2002
Team Italia
3·1,609 Posts 
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 trialfactoring 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 trialfactoring division in 10^100 second. So, altogether the universe could perform 10^300 trialfactoring 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 trialfactoring 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 trialfactoring 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 < 2^1400, and so N < 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 trialfactor 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 trialfactoring 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 trialfactored 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 trialfactoring assignments are greater than M21000000 = 2^21000000  1 which is far, far larger than 2^2800. 8<8<8< Hope this helps... Luigi 
20031124, 17:59  #3 
Mar 2003
Braunschweig, Germany
342_{8} Posts 
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 mathspeculation: 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 nullvalues of the Zetafunction really are linked to quantum dynamical systems (see the MontgomeryOdlyzko 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://wwwusers.cs.york.ac.uk/~susan/cyc/b/big.htm for _really_ big numbers Tau 
20031124, 18:42  #4  
Aug 2002
3×83 Posts 
(much snippage)
Quote:
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. 

20031124, 21:25  #5 
Banned
"Luigi"
Aug 2002
Team Italia
4827_{10} Posts 
Hey, waitaminnit! :P
I only posted Cheesehead message, he's the guru, not me :) Luigi 
20031124, 23:50  #6 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
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 20031124 at 23:57 
20031125, 07:03  #7 
Aug 2002
2^{6}×5 Posts 
A lower upperbound could probably be obtained by considering themodynamic arguments.

20040816, 22:10  #8 
5,233 Posts 
u huh
i believe there is no largest number they just go on and on.

20040817, 02:04  #9 
16037_{8} Posts 
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.

20040817, 12:06  #10  
Banned
"Luigi"
Aug 2002
Team Italia
3·1,609 Posts 
Quote:
I think this is the modern question to the ancient dicotomy between potential and actual Infinite. Luigi 

20040821, 19:21  #11  
1101111000100_{2} Posts 
ask Hawking
Quote:


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