[ExI] 2^57885161-1

[Tomasz Rola]

On Fri, 8 Feb 2013, Dave Sill wrote:

> What's the mathematical importance of 2^257885161-1? Or Mersenne
> primes in general?
> 
> -Dave
> _______________________________________________

Eh, well. With primes in general, one can:

- make strangely quasi regular pictures, like Ulam's spiral

http://io9.com/5818900/the-bizarre-mathematical-conundrum-of-ulams-spiral

(I guess there can be more of them pictures, I just have to find a way to 
do them, heh)

- goedelize ones language - i.e. convert any sequence of symbols into 
unique natural number. In very simple example, my name can be goedelized 
with this primitive code:

(defun goedelize-simple (syms)
  (let ((alf " ABCDEFGHIJKLMNOPQRSTUVWXYZ")
        (goed 1)
        (primes '(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67)))
    (map nil #'(lambda (z)
                 (let* ((posit (position z alf))
                        (prime (car primes))
                        (part (expt prime posit)))
                   (format t " pos(~A)=~A, ~A^~A=~A~%" z posit prime posit 
part)
                   (setf goed (* goed part))
                   (setf primes (cdr primes)))) syms)
    (format t " G(~A)=~A~%" syms goed)))

And after entering it into Common Lisp, one can get this:

CL-USER> (goedelize-simple "TOMASZ")
 pos(T)=20, 2^20=1048576
 pos(O)=15, 3^15=14348907
 pos(M)=13, 5^13=1220703125
 pos(A)=1, 7^1=7
 pos(S)=19, 11^19=61159090448414546291
 pos(Z)=26, 13^26=91733330193268616658399616009
 G(TOMASZ)=721298465788875811087189330837508329315168589988035637799680000000000000

(It is primitive, because it should use some kind of primes generator 
rather than fixed list with twenty-some ready numbers, so usefulness of 
this function is very limited to rather short messages - could be easily 
improved by extending the 'primes list - but version with full blown prime 
generation is unreadable for the expected reader of this email).

In more sophisticated use, Goedel used similar technique in proof of one 
of his famous theorem.

Obviously, one can easily learn my name by checking how many times 
G(TOMASZ) can be divided without the rest by consecutive primes, like, it 
divides 20 times by 2, so first letter is twentieth in alphabet (T), it 
divides 15 times by 3, hence second letter is O and so on.

Goedelized messages had been used by crew of one fictional s-f-spaceship, 
sent with mission to nonexistant planetary system (if I remember well), 
who once freed from civilisation boundaries, quickly developed themselves 
into superhumans, and started sending back home things like "we are coming 
back to whip your arses, get ready you bastards". Sounds to me like 
perfect way to use prime numbers. Unfortunately I keep forgetting the 
title and author, this must be a civilisation-induced block.

- like above, only differently, any natural number is either a prime, or 
can be constructed by multiplying certain primes

- primes are the right thing to use if one wants to make pseudo random 
generator

- primes are handy in some algorithms, for example when implementing hash 
tables

- last but not least, one can use primes in cryptography, secure 
communication, secure banking, setting up artificial currencies - provided 
that every interested party actually wants them secure.

Somehow I don't recall any other good use for primes. From what I have 
read, number theorists for many decades were proud that their field has no 
practical (military etc) applications. I would say, we are just starting 
to learn what to do with primes, which is why the list above is so short. 
Like Anders wrote, they are collectibles, especially the big ones. For 
now.

Regards,
Tomasz Rola

--
** A C programmer asked whether computer had Buddha's nature.      **
** As the answer, master did "rm -rif" on the programmer's home    **
** directory. And then the C programmer became enlightened...      **
**                                                                 **
** Tomasz Rola          mailto:tomasz_rola at bigfoot.com             **