has 13-million digits, or does it?
The "46th" mersenne prime number is (2**43,112,609)-1
Since 1024 > 1000,
and 10**3 = 1000,
and 2**10 = 1024,
then 2**10 > 10**3, by a little (2.4%).
2**43112609 = (2**10)**4311260.9
(2**10)**4311260.9 > (10**3)**4311260.9 = 10**(3*4311260.9) = 10**(12933782.7)
10**12,933,782 = 1 followed by 12,933,782 zeros.
[(10**.7 = 5.0) so no more digits there.] Not 13 million, but close, and this is a lower bound.
The approximation above uses the log of 1000 (ie: 3) as a lower bound to the log of 1024. As an upper bound, the
log of 1024 is just slightly less than 3.0103, so (10**3.103)**4311260.9 > (2**10)**4311260.9 = 2**43112609
This number has 4311260.9*3.0103 digits or 12,978,188.
Sorry, still not 13 million.
The moral: don't believe everything you read.