Which is greater, A=2^79,641,170,620,168,673,833 or B=3^50,247,984,153,525,417,450?

And what about C=2^2^2^83 and D=3^3^3^52?

log(2^2^2^83) = log(2) * 2^2^83

log(log(2^2^2^83)) = log(log(2)) + log(2^2^83)
                              = log(log(2)) + log(2)*(2^83)

and similarly for 3^3^3^52.

list
   10   print log(log(2))+log(2)*2^83
   20   print log(log(3))+log(3)*3^52
OK
run
 6703708186976009930559261.245793649984802979419168522216407338034187576294
 7098223961595389530659098.104816042249321895633570347190098735087988799134
OK

D is bigger than C.

To check how many digits of these two answers are unreliable,

?2^83
 9671406556917033397649408
OK
?3^52
 6461081889226673298932241
OK

so take away about as many digits as make up these numbers away from the end of the results. D is still comfortably larger than C.

Posted by Charlie on 2006-09-22 14:26:48