How many digits are there in 2^1000 (2 to the power of 1000)?

2^0 = 1
2^10 = 1,024 ~ 1,000 (1k)
2^20 = 1,048,576 ~ 1,000,000 (1M)

In this series the number of digits grows by 3 for every tenth power of 2. Therefore a first guess at the number of digits in 2^1000 is 300 (=3*1000/10).

However, the percentage difference between 2^(10x) and 10^(3x) increases as x increases...

(2^0)/(10^0)=1
(2^10)/(10^3)=1.024 ie 2.4% increase
(2^20)/(10^6)=1.049 ie 4.9% increase

I guess (although I can't prove it) that this incease is sufficiently significant for 2^1000 to add a couple of digits to the number.

So my guess is 302 digits (ie 300 plus a couple)

Hopefully somebody has some scientific way of doing this...

Posted by fwaff on 2003-02-20 03:28:44