20090906, 19:17  #1 
Random Account
Aug 2009
19×101 Posts 
Number Of Digits; I Hate To Ask
Is there a formula for calculating the number of digits in a large number?
An example would be 2^{18477} 
20090906, 19:35  #2 
Jul 2003
So Cal
2·3·347 Posts 
Yes, take the base10 log, and round up.
log (2^18477) = 18477 log 2 = 18477 (0.30103) = 5562.1 so 2^18477 has 5563 digits. 
20090906, 19:54  #3 
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Aug 2009
77F_{16} Posts 

20090906, 20:15  #4  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17×251 Posts 
Quote:
The "LOG" button on your calculator is (almost certainly, anyway  just try log(1000) or something to check) the base 10 logarithm. I doubt you'll be able to replicate the first step on your calculator, since if it's anything like my TI, it has a limit of 100 digits. You can, however, replicate the rest. Note that 0.30103 is not exactly log_10(2), but is close enough for this example. If you did need the base x logarithm of y, you could use log(y)/log(x). (with log being the logarithm of any base, surprising as that may sound ) 

20090906, 20:31  #5 
Random Account
Aug 2009
3577_{8} Posts 
On my calculator if I enter 1000 and press Log, it returns 3, meaning 10^{3}
Entering 2 and pressing Log gives me .301029996. This is the rounded number in his example. Multiply that times 18477 and i get 5562.1. It seems to work. On the GIMPS home page is 2^{42643801}. If I do this like above, then I get 12,837,063.2 
20090906, 20:35  #6 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17×251 Posts 
Yep, you've got it.

20090906, 21:52  #7 
Aug 2004
Melbourne, Australia
230_{8} Posts 

20090906, 22:09  #8 
Aug 2006
1011101100001_{2} Posts 
Yes, the 'right' way is to round down and add one. But usually you'd know if you were dealing with a power of 10.

20090906, 23:24  #9 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
Hmmm... this should be on the Mersenne Wiki.

20090907, 00:16  #10 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10010010101101_{2} Posts 

20090907, 01:23  #11 
Random Account
Aug 2009
19×101 Posts 
http://www.mersenne.org/bench.htm
This link is broken. It is in the last post of the thread above. Understandable. All of this is from 2004. 
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