rec.puzzles Archive

What digits does googol!

arithmetic/digits/most.significant/googol

Problem

What digits does googol! start with?
Show solution
I'm not sure how to calculate the first googol of digits of log10(e), but
here's the first 150(approximately) of them...

0.43429448190325182765112891891660508229439700580366656611445378316586464920
8870774729224949338431748318706106744766303733641679287158963906569221064663

We need to deal with the digits immediately after the decimal point in
googol*log10(e), which are  .187061

frac[log(googol!)] = frac[halflog2pi + 50 + googol(100-log10(e))]
 = frac{halflog2pi + frac[googol(100-log10(e))]}
 = frac[.399090 + (1- .187061)]
 = .212029

10 ** .212029 = 1.629405

Which means that googol! starts with 1629