[extropy-chat] damien's psi book

[spike]

> Subject: Re: [extropy-chat] damien's psi book
> 
> On Wed, 9 Feb 2005 21:08:18 -0800, spike <spike66 at comcast.net> wrote:
> >
> > ...  But what about the tens column: are
> > the tens column of the primes evenly distributed
> > enough over all ten digits to pass the traditional
> > tests for randomness?

Four extropians have pinged me offlist, so I will answer
here: the lower digits are slightly more common than the 
higher digits.  There are slightly more 1s than 9s.

> >  How do you prove it?

Heres the arm-waving proof:

You know that the number of primes less than n is
approximated by the function n/ln(n).  Focus for the
moment on seven digit numbers.  Use that natural log
function to approximate that there are about 72400
primes less than 1000000 and about 137800 primes
less than 2000000 so there are about 65400 seven
digit primes with the leading digit 1.  

Use the same reasoning to approximate that there are
about 58700 seven digit primes with the leading digit 
of 9.  So there are about 11 percent more 1s than 9s 
in the leading digit.

I knew this back so many years in college, but
here's the kicker that I missed before: you can
use that same line of reasoning and find that there
is an access one tenth as high in the next digit
to the right: there is about 1.1 percent more 1s 
than 9s in the hundred-thousands column, and 
likewise about 0.11 percent more 1s than 9s
in the next column, and so on.  So technically one
could argue the distribution of digits in the 
primes cannot be expected to pass the tests of
randomness.

{8-]

I generated all the primes less than a billion
to test the theory.  It works.

Ain't that cool?  {8^D    

ps Next time I will post some thoughts on what all
this has to do with the lottery, and the limits
of our ability to generate true random numbers,
yea verily, even the limits of our ability to
generate truly random numbers using mechanical 
devices such as a slot machine or those lottery 
number generating ping pong ball thingies.

spike