mac * maths * Prime Thoughts...
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(Thanks and 42 flips of the towel to Carl Sagan
 see note below as to why)...

There are certain questions concerning the randomness of 
primes (or at least their apparant distributions)...

The density of prime-twins,

Also, the density of primes that end in say "23" or "51" or "97"
For each block (eg, 1-10k,  10k-20k, 20k-30k  etc 
is there a "more probably last digits set".

And as we go further out, we could look at the distribution of
the last 3-digits of a prime.  These again lead back to  the
randomness of prime distributions.

One almost wants to ask questions of a "Monte Carlo" nature
about the distribution of primes, and the digits "patterns"
in pi.

Is there a prime expression for pi (other than the
classic one), and why is it a "reciprocal" relationship.
The division function "recapitulates" itself in the
natures of divisibility (prime-ness) and then again
in terms of the "pi-ness" of things?   All very odd.

And of course the "distribution" of ending-digits 
is based on the fact that we are using BASE-10 --
that's the reason why there won't ever be any last
digits of primes like:

05 15 25 35 45 65 75 85 95 <-- these occur because the base 
                               has "5" as a factor    
10 20 30 40 50 60 70 80 90 <-- these occur because of the base

This begs the question of whether there is a "natural"
base that should be used.  That is:  Is there something
about numbers that shows that there is a "prefered"
base?  In theory, it should turn out that the answer is
no of course.

Consider expressing numbers in BASE 3, then by the 
above list and a "similarity" argument, we should expect
that we would not ever see any primes ending (in base 3)
        10 20 

The primes in base 3 are:
     2   3   5   7   11  13  17  19   23   29   31
    02  10  12  21  102 111 122 201  212 1002 1011
Note the rare occurence of two consecutive primes that
are both "even" (ie, end in 2 -- but "even-ness" doesn't
mean in base 3 what it does in base 10!!!

   Thus, while it looks like "212" (base 3 for "23")
   is a multiple of 2 ("2" in base 10), the division
   reveals this (is this complicated enough??? ;)
   Performing the division in BASE 3....

             1 0 2 
       2 |   2 1 2
           - 2
             0 1 2
           -   1 1  
                 1 <-- remainder of 1,
                       there-fore 212 (base 3) 
                       is NOT a multiple of 2.

The pattern is clear:  No primes greater than "3" (103)
will ever end with a zero.  (Otherwise they'd be a
factor of 3, just like n*10 ends with a zero in base 10).

In general, the primes tend to alternate between ending
in 1 and then 2.  Thus, we seek (in addition to 
twin-primes), occurences of consecutive primes that
end in the same digit.

It "seems" like the patterns in one base may (or may not
as Broomfondle might well say), mirror the patterns in
another base. 

Of course, all of these questions are just as "mysterious"
as the yearnings to find the "utlimate" volume in
the Library of Babel.

Is there even any point in asking such questions?


btw:  I can not take complete credit for this idea of
      their being "patterns" in the various bases.
In his novel, "Contact", Carl Sagan
discusses that the universe was "constructed" and as such
the number pi (out about a billion places or so),
when displayed in base 13, displays a very NON-random
pattern:  A string of ones and zeroes that end up
forming a picture of sorts.  

(Of course, it's left to the imagination of the reader
as to whether this means that there IS a god (who
designed pi and hence the universe -- which came
first pi or the universe?), or that it's just
a "coincidence" -- that is, if you keep trying
combinations of BASE's and look long enough into
the value of pi then sooner or later you'd find
SOME pattern).

So, maybe the search for pi (primes?) are the 
search for the divine, for some "deeper meaning"
in the structure of the universe -- or at least
in the "mathematical universe".   So, instead
of asking ourselves how many angels can dance 
on the head of a pin (the answer of course being
"as many as want to"), we now ask ourselves,
how many digits of pi does it take to make a rose?

or a DNA molecule (even with our without a towel)....                      

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